Series and Progressions

Overview

The systematic study of series and progressions constitutes a fundamental pillar of quantitative reasoning. These concepts govern the principles of ordered sets of numbers, where each term is related to the preceding one by a specific, discernible rule. A mastery of this topic is not merely an exercise in formulaic application; rather, it cultivates the analytical ability to recognize patterns, extrapolate logical sequences, and structure complex problems into manageable components. For the GATE examination, proficiency in this area is indispensable, as questions frequently test a candidate's capacity for logical deduction and swift pattern identification under timed conditions.

In this chapter, we shall explore the foundational types of progressions: Arithmetic and Geometric. We will establish the formal definitions and derive the essential formulae for determining any term in a sequence and the sum of its elements. Beyond these structured progressions, we will also investigate the broader category of number series, where the underlying patterns may be more varied and complex, demanding a higher degree of logical acuity. Our objective is to equip you with a methodical approach to deconstruct these problems, enabling you to solve them with both accuracy and efficiency.

---

Chapter Contents

| # | Topic | What You'll Learn |
|---|-------|-------------------|
| 1 | Arithmetic Progression (AP) | Understanding sequences with a constant difference. |
| 2 | Geometric Progression (GP) | Analyzing sequences with a constant ratio. |
| 3 | Number Series | Identifying patterns in logical number sequences. |

---

Learning Objectives

---

We now turn our attention to Arithmetic Progression (AP)...

Part 1: Arithmetic Progression (AP)

Introduction

In our study of numerical computation, we frequently encounter sequences of numbers that follow a discernible pattern. Among the most fundamental of these is the Arithmetic Progression (AP). An arithmetic progression is a sequence where the difference between any two consecutive terms remains constant. This simple, elegant structure allows for powerful generalizations regarding its terms and their sums.

A thorough understanding of arithmetic progressions is essential for the GATE examination, as it forms the bedrock for solving a variety of problems related to series, financial calculations, and pattern recognition. The principles we establish here are not merely abstract mathematical concepts; they are tools for efficiently analyzing and manipulating ordered numerical data. This chapter will formally define an arithmetic progression, derive the formulae for its general term and the sum of its terms, and explore key properties and problem-solving techniques relevant to the examination.

---

---

---

Key Concepts

1. The General Term of an AP

Let us consider an arithmetic progression with the first term $a$ and a common difference $d$. The terms of the sequence can be expressed as follows:

• The first term is $a_1 = a$.
• The second term is $a_2 = a + d$.
• The third term is $a_3 = a_2 + d = (a+d) + d = a + 2d$.
• The fourth term is $a_4 = a_3 + d = (a+2d) + d = a + 3d$.

Observing the pattern, we can see that the $n^{th}$ term of the AP is obtained by adding the common difference $(n-1)$ times to the first term. This gives us the general formula for the $n^{th}$ term.

Worked Example:

Problem: Find the 25th term of the arithmetic progression: $3, 8, 13, 18, \dots$.

Solution:

Step 1: Identify the first term ($a$) and the common difference ($d$).

The given sequence is $3, 8, 13, 18, \dots$.
The first term is $a = 3$.
The common difference is $d = a_2 - a_1 = 8 - 3 = 5$.

Step 2: Identify the term number ($n$).

We need to find the 25th term, so $n = 25$.

Step 3: Apply the formula for the $n^{th}$ term, $a_n = a + (n-1)d$.

Step 4: Perform the calculation.

Answer: The 25th term of the given AP is $123$.

---

2. The Sum of the First $n$ Terms of an AP

A frequent requirement in problems involving arithmetic progressions is to compute the sum of a certain number of its terms. Let $S_n$ denote the sum of the first $n$ terms of an AP with first term $a$ and common difference $d$.

We can write the sum in two ways:

Adding these two equations term by term, we find that each pair of corresponding terms sums to $2a + (n-1)d$. Since there are $n$ such pairs, we have:

This leads to the primary formula for the sum of an AP.

We can also express this formula in a different form. If we denote the last term, $a_n$, by $l$, then $l = a + (n-1)d$. Substituting this into the sum formula:

This alternative formula is particularly useful when the first and last terms are known.

Worked Example:

Problem: Find the sum of the first 40 terms of the AP: $10, 7, 4, 1, \dots$.

Solution:

Step 1: Identify the key parameters from the given AP.

First term, $a = 10$.
Common difference, $d = 7 - 10 = -3$.
Number of terms, $n = 40$.

Step 2: Apply the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$.

Step 3: Simplify the expression inside the brackets.

Step 4: Compute the final sum.

Answer: The sum of the first 40 terms is $-1940$.

---

3. Properties of Arithmetic Progression

Arithmetic progressions exhibit several useful properties that can simplify problem-solving.

Effect of a Constant: If a constant value is added to, subtracted from, multiplied by, or divided into (for non-zero constants) every term of an AP, the resulting sequence is also an AP.

Sum of Equidistant Terms: In a finite AP, the sum of any two terms equidistant from the beginning and the end is constant and is equal to the sum of the first and last terms. That is, $a_k + a_{n-k+1} = a_1 + a_n$.

Arithmetic Mean: Three numbers $x, y, z$ are in AP if and only if $2y = x + z$. The term $y$ is called the arithmetic mean of $x$ and $z$.

Sum of First $n$ Natural Numbers: The sum of the first $n$ natural numbers, $1+2+3+\dots+n$, is an AP with $a=1, d=1$. The sum is given by $S_n = \frac{n(n+1)}{2}$.

Sum of First $n$ Positive Odd Numbers: The sequence of positive odd numbers is $1, 3, 5, \dots$. This is an AP with $a=1, d=2$. The sum of the first $n$ terms is $S_n = n^2$.

Sum of First $n$ Positive Even Numbers: The sequence of positive even numbers is $2, 4, 6, \dots$. This is an AP with $a=2, d=2$. The sum of the first $n$ terms is $S_n = n(n+1)$.

---

Problem-Solving Strategies

---

---

Common Mistakes

---

Practice Questions

:::question type="MCQ" question="The sum of the first 30 consecutive positive even integers is divided by the sum of the first 30 consecutive positive integers. The result is:" options=["2","30","$\frac{61}{31}$","$\frac{62}{31}$"] answer="2" hint="Use the direct formulas for the sum of first n even integers and first n natural numbers." solution="
Step 1: Calculate the sum of the first 30 consecutive positive even integers.
The sum of the first $n$ positive even integers is given by the formula $S_{even} = n(n+1)$.
For $n=30$, we have:

Step 2: Calculate the sum of the first 30 consecutive positive integers.
The sum of the first $n$ positive integers is given by the formula $S_{natural} = \frac{n(n+1)}{2}$.
For $n=30$, we have:

Step 3: Divide the sum from Step 1 by the sum from Step 2.

Step 4: Simplify the fraction.

Answer: \boxed{2}
"
:::

:::question type="NAT" question="The 7th term of an Arithmetic Progression is 22 and the 12th term is 37. What is the value of the first term?" answer="4" hint="Set up two linear equations using the formula $a_n = a + (n-1)d$ and solve for $a$ and $d$." solution="
Step 1: Formulate equations based on the given information.
Let the first term be $a$ and the common difference be $d$.
The $n^{th}$ term is given by $a_n = a + (n-1)d$.

For the 7th term ($n=7$):

For the 12th term ($n=12$):

Step 2: Solve the system of linear equations. Subtract equation (1) from equation (2).

Step 3: Substitute the value of $d$ back into equation (1) to find $a$.

Answer: \boxed{4}
"
:::

:::question type="MSQ" question="A sequence is defined by $a_n = 5n - 3$ for $n \ge 1$. Which of the following statements is/are true?" options=["The sequence is an Arithmetic Progression.","The common difference is -3.","The sum of the first 10 terms is 245.","The 100th term is 497."] answer="The sequence is an Arithmetic Progression.,The sum of the first 10 terms is 245.,The 100th term is 497." hint="Check if the difference $a_{n+1} - a_n$ is a constant. Then, calculate the specific values requested in the options using the appropriate AP formulas." solution="
Let us analyze the given sequence $a_n = 5n - 3$.

Option A: The sequence is an Arithmetic Progression.
To check this, we find the difference between consecutive terms:

Since the difference is a constant (5), the sequence is an AP. Thus, this statement is true.

Option B: The common difference is -3.
From our calculation above, the common difference $d = 5$. Thus, this statement is false.

Option C: The sum of the first 10 terms is 245.
First, we find the first term ($a_1$) and the 10th term ($a_{10}$).

Using the sum formula $S_n = \frac{n}{2}(a_1 + a_n)$:

Thus, this statement is true.

Option D: The 100th term is 497.
We use the given formula for $a_n$ with $n=100$:

Thus, this statement is true.

Answer: \boxed{\text{A, C, and D}}
"
:::

:::question type="NAT" question="How many integers between 100 and 400 are divisible by 7?" answer="43" hint="The numbers divisible by 7 form an AP. Find the first term ($a$) and the last term ($l$) in the given range. Then use the formula $l = a + (n-1)d$ to find $n$." solution="
Step 1: Identify the sequence.
The integers between 100 and 400 that are divisible by 7 form an Arithmetic Progression with a common difference $d=7$.

Step 2: Find the first term ($a$) of this AP.
We need the smallest integer greater than or equal to 100 that is a multiple of 7.
Dividing 100 by 7 gives $100 = 7 \times 14 + 2$.
The next multiple of 7 is $7 \times 15 = 105$. So, $a = 105$.

Step 3: Find the last term ($l$ or $a_n$) of this AP.
We need the largest integer less than or equal to 400 that is a multiple of 7.
Dividing 400 by 7 gives $400 = 7 \times 57 + 1$.
The multiple of 7 just before 400 is $7 \times 57 = 399$. So, $l = 399$.

Step 4: Use the formula for the $n^{th}$ term to find the number of terms, $n$.

Answer: \boxed{43}
"
:::

---

Summary

---

What's Next?

---

---

Part 2: Geometric Progression (GP)

Introduction

In our study of numerical computation, we frequently encounter sequences of numbers that follow a discernible pattern. Among the most fundamental of these are progressions. While an arithmetic progression is characterized by a common difference between successive terms, a geometric progression is defined by a common ratio. This structure arises naturally in various applications, including compound interest calculations, population growth models, and the analysis of algorithms where a problem's size is reduced by a constant factor at each step.

A thorough understanding of geometric progressions is essential for developing the mathematical maturity required for quantitative aptitude sections in examinations like GATE. We will explore the core properties of a GP: the formula for the $n^{\text{th}}$ term, the sum of a finite number of terms, and the condition under which an infinite geometric series converges to a finite sum. Mastery of these concepts provides a powerful tool for solving a specific class of problems efficiently and accurately.

---

Key Concepts

The analysis of a geometric progression primarily revolves around three fundamental calculations: finding a specific term, finding the sum of a finite number of terms, and finding the sum of an infinite series.

1. The $n^{\text{th}}$ Term of a GP

To find any specific term in a geometric progression, we need only the first term and the common ratio. We observe that the $k^{\text{th}}$ term in the sequence has a power of $r$ equal to $k-1$. This pattern leads directly to the general formula for the $n^{\text{th}}$ term.

Worked Example:

Problem: Find the 8th term of the geometric progression: $3, 6, 12, 24, \dots$

Solution:

Step 1: Identify the first term ($a$) and the common ratio ($r$).
The first term is $a=3$. To find the common ratio, we divide any term by its preceding term.

We can verify this with the next pair: $r = \frac{12}{6} = 2$.

Step 2: Apply the formula for the $n^{\text{th}}$ term, $T_n = ar^{n-1}$, with $n=8$.

Step 3: Simplify the expression.

Step 4: Compute the final value. We know that $2^7 = 128$.

Answer: The 8th term of the series is $384$.

---

2. Sum of the First $n$ Terms of a GP

A frequent requirement in problems involving progressions is to calculate the sum of a certain number of terms. For a GP, we can derive a concise formula for this sum, which avoids the need to manually add all the terms.

Let the sum of the first $n$ terms be $S_n$.

Multiplying by $r$, we get:

Subtracting the second equation from the first, most terms cancel out, leaving:

This yields the general formula.

Worked Example:

Problem: Find the sum of the first 5 terms of the GP: $5, 15, 45, \dots$

Solution:

Step 1: Identify $a$, $r$, and $n$.
Here, $a=5$, $n=5$. The common ratio is:

Step 2: Since $r > 1$, we use the formula $S_n = \frac{a(r^n-1)}{r-1}$.

Step 3: Calculate the values in the expression.

Step 4: Simplify to find the final sum.

Answer: The sum of the first 5 terms is $605$.

---

3. Sum of an Infinite GP

We now consider the intriguing case of summing an infinite number of terms. It may seem counter-intuitive that an infinite sum could result in a finite value. However, if the terms decrease in magnitude sufficiently quickly, the sum can converge. This occurs when the absolute value of the common ratio is less than 1.

Consider the formula

If $|r| < 1$, then as $n \to \infty$, the term $r^n \to 0$. The formula for the sum simplifies elegantly.

Worked Example:

Problem: Find the sum of the infinite geometric series $16, 8, 4, 2, \dots$

Solution:

Step 1: Identify the first term $a$ and the common ratio $r$.

Step 2: Check if the condition for convergence, $|r| < 1$, is met.
Since $|1/2| < 1$, the series converges and we can find its sum.

Step 3: Apply the infinite sum formula

Step 4: Simplify the expression.

Answer: The sum of the infinite series is $\boxed{32}$.

---

Problem-Solving Strategies

---

Common Mistakes

---

Practice Questions

:::question type="MCQ" question="The 4th term of a geometric progression is 24 and the 7th term is 192. What is the first term?" options=["2","3","4","6"] answer="3" hint="Set up two equations using the formula $T_n = ar^{n-1}$ and solve for 'a' and 'r' by dividing one equation by the other." solution="
Step 1: Write the equations for the 4th and 7th terms.
Let the first term be $a$ and the common ratio be $r$.

Step 2: Divide equation (2) by equation (1) to find $r$.

Step 3: Substitute the value of $r$ back into equation (1) to find $a$.

Result: The first term is $\boxed{3}$.
"
:::

:::question type="NAT" question="A ball is dropped from a height of 81 meters. Each time it bounces, it rises to $\frac{2}{3}$ of the height from which it fell. Calculate the total distance (in meters) traveled by the ball before it comes to rest." answer="405" hint="The total distance is the initial drop height plus the sum of an infinite series representing the up-and-down travel during the bounces." solution="
Step 1: Analyze the distance traveled.
The total distance is the sum of the downward distances and upward distances.
Initial drop:

Step 2: Calculate the distances for subsequent bounces.
Height of 1st bounce (upward):

Distance of 1st bounce (up + down):
Height of 2nd bounce (upward):
Distance of 2nd bounce (up + down):

The total distance is $81 + (108 + 72 + 48 + \dots)$.
The part in the parenthesis is a GP.

Step 3: Identify the GP for the bounce distances.
The series of bounce distances is $108, 72, 48, \dots$.
First term

Common ratio

Step 4: Calculate the sum of this infinite GP.
Since $|r| = \frac{2}{3} < 1$, the sum converges.

Step 5: Calculate the total distance.

Result: The total distance traveled is $\boxed{405}$ meters.
"
:::

:::question type="MSQ" question="Consider a geometric progression with first term $a=2$ and common ratio $r = -2$. Which of the following statements is/are correct?" options=["The 5th term is 32.","The sum of the first 4 terms is -10.","The sum to infinity exists and is finite.","The terms of the sequence alternate in sign."] answer="The 5th term is 32.,The sum of the first 4 terms is -10.,The terms of the sequence alternate in sign." hint="Evaluate each statement independently using the standard GP formulas. Pay close attention to the negative common ratio." solution="
Statement A: The 5th term is 32.

This statement is correct.

Statement B: The sum of the first 4 terms is -10.

This statement is correct.

Statement C: The sum to infinity exists and is finite.
The condition for a finite sum to infinity is $|r| < 1$. Here, $r = -2$, so $|r| = 2$. Since $|r| \ge 1$, the series diverges and the sum to infinity does not exist as a finite value.
This statement is incorrect.

Statement D: The terms of the sequence alternate in sign.
The sequence is:

A negative common ratio causes the sign of consecutive terms to alternate.
This statement is correct.

Result: The correct statements are A, B, and D.
"
:::

---

Summary

---

What's Next?

---

---

Part 3: Number Series

Introduction

The study of number series is a fundamental component of quantitative aptitude, designed to assess a candidate's capacity for logical deduction and pattern recognition. In the context of the GATE examination, these problems are not merely about computation but about discerning the underlying structure that governs a sequence of numbers. A series can be governed by a simple arithmetic rule, a more complex recursive relationship, or a combination of several operations.

Our objective in this chapter is to develop a systematic methodology for analyzing these sequences. We will move from foundational concepts such as arithmetic and geometric progressions to more intricate patterns involving differences, powers, and recursive definitions. Mastery of this topic requires not only familiarity with common patterns but also a flexible, analytical approach to problem-solving, which we shall cultivate through rigorous explanation and carefully chosen examples.

---

Key Concepts

1. Standard Progressions

The most elementary forms of series are the arithmetic and geometric progressions, which are defined by a constant difference or a constant ratio between consecutive terms, respectively.

2. Difference-Based Series

Many series that are not simple APs or GPs reveal a pattern in the differences between their consecutive terms. This "method of differences" is a powerful analytical tool.

Consider a sequence $T_1, T_2, T_3, \dots$. We can form a new sequence of first differences, $\Delta_1$, where the terms are $(T_2 - T_1), (T_3 - T_2), \dots$. If this new sequence is not an AP or GP, we can take the second differences, $\Delta_2$, which are the differences of the terms in $\Delta_1$. This process can be repeated.

Common Patterns in Differences:

• The first differences form an AP.


• The second differences form an AP.


• The differences are squares, cubes, prime numbers, or follow another identifiable pattern.

Worked Example:

Problem: Find the next term in the series $2, 3, 6, 13, 26, \dots$

Solution:

Step 1: Write down the series and calculate the first differences ($\Delta_1$).

The series is: $2, 3, 6, 13, 26, x$

We observe that the first differences do not form a simple AP.

Step 2: Calculate the second differences ($\Delta_2$) from the first differences.

The first differences are: $1, 3, 7, 13, y$

The second differences form an AP with a common difference of 2.

Step 3: Extrapolate the pattern to find the next terms.

The next term in $\Delta_2$ will be $6 + 2 = 8$.

We use this to find the next term in $\Delta_1$, which we called $y$.

Step 4: Use the extrapolated first difference to find the next term in the original series.

The next term in the original series, $x$, is found by adding the next first difference (21) to the last known term (26).

Answer: The next term in the series is $47$.

3. Recurrence Relations

A sequence can also be defined by a recurrence relation, where each term is a function of one or more preceding terms. Such definitions are common in computational contexts.

A well-known example is the Fibonacci sequence, where $F_n = F_{n-1} + F_{n-2}$. GATE questions may present a table of values and require the deduction of the underlying recurrence relation, which can involve arithmetic, multiplication, or a combination of operations based on previous terms and new inputs.

A critical skill is the ability to work such relations in reverse. If a relation is of the form $T_{n+1} = f(T_n, T_{n-1})$, one can often rearrange it to find $T_{n-1}$ in terms of $T_n$ and $T_{n+1}$.

Worked Example:

Problem: A sequence is defined by the relation $A_{n} = A_{n-1} + A_{n-2}$ for all $n \geq 3$. If $A_7 = 45$ and $A_8 = 73$, what is the value of $A_3$?

Solution:

Step 1: Rearrange the recurrence relation to solve for a preceding term.

Given the relation:

We can isolate the earliest term, $A_{n-2}$:

Step 2: Apply the rearranged formula iteratively to work backwards from the known terms.

To find $A_6$, we set $n=8$:

To find $A_5$, we set $n=7$:

To find $A_4$, we set $n=6$:

Step 3: Compute the final required term, $A_3$.

To find $A_3$, we set $n=5$:

Answer: The value of $A_3$ is $6$.

4. Applications of LCM in Periodic Events

Certain problems that appear alongside number series questions involve determining the time at which multiple periodic events will occur simultaneously. These are fundamentally problems about finding the Least Common Multiple (LCM) of the periods.

Worked Example:

Problem: Three alarms are set to beep at intervals of 15 minutes, 25 minutes, and 35 minutes, respectively. If they all beep together at 8:00 AM, at what time will they next beep together?

Solution:

Step 1: Identify the periods of the individual events.

The periods are $P_1 = 15$ min, $P_2 = 25$ min, and $P_3 = 35$ min.

Step 2: Find the prime factorization of each period.

Step 3: Calculate the LCM by taking the highest power of each prime factor present in the factorizations.

Step 4: Convert the LCM from minutes to hours and minutes to find the future time.

The remainder is $525 \pmod{60} = 45$. The quotient is $\lfloor 525/60 \rfloor = 8$.

Step 5: Add this duration to the initial time.

Initial time = 8:00 AM
Duration = 8 hours 45 minutes

Next beep time = 8:00 AM + 8 hours 45 minutes = 4:45 PM.

Answer: The alarms will next beep together at 4:45 PM.

---

Problem-Solving Strategies

---

Common Mistakes

---

Practice Questions

:::question type="MCQ" question="What is the next term in the sequence $4, 5, 13, 40, 104, \dots$?" options=["229","208","185","244"] answer="229" hint="Calculate the differences between consecutive terms, and then repeat the process on the resulting sequence of differences." solution="
Step 1: Write down the series and find the first differences ($\Delta_1$).
Series: $4, 5, 13, 40, 104, x$

Step 2: Analyze the sequence of first differences.
The sequence of differences is $1, 8, 27, 64$. We recognize these as the cubes of natural numbers.

Step 3: Determine the next term in the difference sequence.
The next term will be $5^3$.

Step 4: Add this next difference to the last term of the original series to find the unknown term $x$.

Result: The next term is 229.
"
:::

:::question type="NAT" question="A sequence is defined by the recurrence relation $T_{n+1} = T_n + 2T_{n-1}$ for $n \ge 2$. If $T_5 = 55$ and $T_4 = 21$, what is the value of $T_1$?" answer="2" hint="Rearrange the recurrence relation to solve for $T_{n-1}$ in terms of $T_{n+1}$ and $T_n$. Then, work backwards from the given terms." solution="
Step 1: Rearrange the recurrence relation to solve for the preceding term, $T_{n-1}$.
Given: $T_{n+1} = T_n + 2T_{n-1}$

Step 2: Use the rearranged formula to find $T_3$. Let $n=4$.

Step 3: Use the formula to find $T_2$. Let $n=3$.

Step 4: Use the formula to find $T_1$. Let $n=2$.

Answer: \boxed{7.5}
"
:::

:::question type="MSQ" question="Consider the sequence $1, 4, 27, 16, 125, 36, \dots$. Which of the following statements is/are correct?" options=["The sequence is an interleaving of two different sequences.","The next term in the sequence is 343.","The terms at odd positions are perfect squares.","The terms at even positions are perfect squares."] answer="The sequence is an interleaving of two different sequences.,The next term in the sequence is 343.,The terms at even positions are perfect squares." hint="Examine the terms at odd positions and even positions separately." solution="
Let's analyze the sequence by splitting it into two interleaved sub-sequences.

Sub-sequence 1 (Odd positions): $1, 27, 125, \dots$
These terms are:
$1 = 1^3$
$27 = 3^3$
$125 = 5^3$
This is a sequence of cubes of odd numbers. The next term would be $7^3 = 343$.

Sub-sequence 2 (Even positions): $4, 16, 36, \dots$
These terms are:
$4 = 2^2$
$16 = 4^2$
$36 = 6^2$
This is a sequence of squares of even numbers.

Now let's evaluate the options:

• \"The sequence is an interleaving of two different sequences.\" This is correct. We have identified two separate patterns for odd and even positions.


• \"The next term in the sequence is 343.\" The last given term, 36, is at the 6th (even) position. The next term will be at the 7th (odd) position. As determined from sub-sequence 1, this term is $7^3 = 343$. This statement is correct.


• \"The terms at odd positions are perfect squares.\" This is incorrect. The terms at odd positions ($1, 27, 125$) are perfect cubes.


• \"The terms at even positions are perfect squares.\" This is correct. The terms at even positions ($4, 16, 36$) are perfect squares.

Therefore, the correct options are the first, second, and fourth statements.
"
:::

:::question type="MCQ" question="Two automated systems, A and B, start a task at 9:00 AM. System A completes a cycle every 45 seconds, and system B completes a cycle every 75 seconds. After starting simultaneously, when will they next complete a cycle at the same instant?" options=["9:03:45 AM","9:03:15 AM","9:04:00 AM","9:02:45 AM"] answer="9:03:45 AM" hint="This problem requires finding the Least Common Multiple (LCM) of the two cycle times." solution="
Step 1: Identify the periods of the two systems.
Period of A, $P_A = 45$ seconds.
Period of B, $P_B = 75$ seconds.

Step 2: Find the prime factorization of each period.

Step 3: Calculate the LCM of the two periods.
The LCM is found by taking the highest power of each prime factor.

Step 4: Convert the total seconds into minutes and seconds.

Step 5: Add this duration to the starting time.
Start time: 9:00:00 AM
Duration: 3 minutes 45 seconds
Next simultaneous completion time: 9:03:45 AM.

Answer: \boxed{9:03:45 \text{ AM}}
"
:::

---

Summary

---

What's Next?

---

Chapter Summary

---

Chapter Review Questions

:::question type="MCQ" question="The first, third, and seventh terms of an arithmetic progression (AP) are in geometric progression (GP). If the first term of the AP is non-zero, what is the common ratio of the GP?" options=["1","2","1/2","4"] answer="B" hint="Let the AP be defined by its first term $a$ and common difference $d$. Express the required terms of the AP and apply the condition for three terms to be in a GP." solution="Let the arithmetic progression be denoted by its first term $a$ and common difference $d$.
The terms of the AP are $a, a+d, a+2d, \dots$.

The first term is:

The third term is:

The seventh term is:

It is given that these three terms, $T_1, T_3, T_7$, are in a geometric progression. For any three terms $x, y, z$ to be in GP, the condition $y^2 = xz$ must hold.
Applying this condition, we have:

Expanding the terms:

Subtracting $a^2$ from both sides:

Rearranging the terms to solve for the relationship between $a$ and $d$:


Since the terms form a progression, we can assume the common difference $d \neq 0$. We can divide both sides by $2d$:

Now, we can find the terms of the GP by substituting $a = 2d$:

• First term of GP:

• Second term of GP:

• Third term of GP:

The geometric progression is $2d, 4d, 8d, \dots$.
The common ratio, $r$, is the ratio of any term to its preceding term:

Thus, the common ratio of the GP is 2.
Answer: \boxed{2}"
:::

:::question type="NAT" question="A ball is dropped from a height of 81 meters. Every time it hits the ground, it rebounds to $\frac{2}{3}$ of the height from which it fell. The total distance, in meters, traveled by the ball before it comes to rest is:" answer="405" hint="The total distance is the initial downward journey plus the sum of all subsequent upward and downward journeys. These subsequent journeys form an infinite geometric series." solution="The problem can be broken down into two parts: the initial drop and the subsequent bounces.

Initial Drop: The ball first travels a distance of 81 meters downwards.

Subsequent Bounces:

- First bounce: The ball rebounds to a height of:

It travels 54 m up and 54 m down. Total distance for this bounce is:

- Second bounce: It rebounds to a height of:

The distance traveled is:

- Third bounce: It rebounds to a height of:

The distance traveled is:

The total distance traveled during the bounces forms an infinite geometric series: $108, 72, 48, \dots$.
The first term of this series is:

The common ratio is:

Since $|r| = \frac{2}{3} < 1$, the sum of this infinite series converges. The sum is given by the formula $S_\infty = \frac{a}{1-r}$.

The total distance traveled by the ball is the sum of the initial drop and the total distance from all the bounces.

The total distance is 405 meters.
Answer: \boxed{405}"
:::

:::question type="MCQ" question="What is the next term in the sequence: 2, 5, 11, 23, 47, ...?" options=["94","95","96","97"] answer="B" hint="Analyze the difference between consecutive terms, or look for a multiplicative and additive relationship between terms of the form

" solution="We can solve this problem by identifying the pattern in the series. There are two primary methods.

Method 1: Analyzing the Differences
Let us find the difference between consecutive terms:

• $$5 - 2 = 3$$


• $$11 - 5 = 6$$


• $$23 - 11 = 12$$


• $$47 - 23 = 24$$

The differences are $3, 6, 12, 24$. This sequence of differences is a geometric progression with a common ratio of $r=2$.
The next difference in this sequence will be:

Therefore, the next term in the original sequence will be the previous term plus this difference:

Method 2: Finding a Recurrence Relation
Let us examine the relationship between a term and the next.

• To get from 2 to 5:
  $$2 \times 2 + 1 = 5$$


• To get from 5 to 11:
  $$5 \times 2 + 1 = 11$$


• To get from 11 to 23:
  $$11 \times 2 + 1 = 23$$


• To get from 23 to 47:
  $$23 \times 2 + 1 = 47$$

The pattern is consistent. Each term is obtained by multiplying the previous term by 2 and adding 1. The recurrence relation is:

Applying this rule to find the next term after 47:

Both methods yield the same result. The correct answer is 95.
Answer: \boxed{95}"
:::

:::question type="NAT" question="The sum of the first $n$ terms of a sequence is given by the formula $S_n = 3n^2 + 5n$. Find the value of the 10th term of this sequence." answer="62" hint="The $n$-th term of a sequence can be found by calculating the difference between the sum of the first $n$ terms and the sum of the first $(n-1)$ terms, i.e.,

" solution="We are given the formula for the sum of the first $n$ terms:

The $n$-th term, $a_n$, is the difference between the sum of the first $n$ terms and the sum of the first $(n-1)$ terms.

First, let's write the expression for $S_{n-1}$ by substituting $(n-1)$ in place of $n$:




Now, we can find $a_n$:



This is the general formula for the $n$-th term of the sequence. To find the 10th term, we substitute $n=10$:

The value of the 10th term is 62.
Answer: \boxed{62}"
:::

---

What's Next?