Sequences, Progressions and Patterns Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Sequences, Progressions and Patterns notes for XAT: 11 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Sequences, Progressions and Patterns

Chapter Roadmap

Step 1: Foundations
Define terms, standard notation, and the core difference between an ordered list and a cumulative sum.

Step 2: Arithmetic Progressions
Master the common difference, nth term, sum formulas, and symmetric properties of A.P.

Step 3: Geometric Progressions
Understand the common ratio, finite sums, and the critical concept of infinite geometric series.

Step 4: Harmonic & Special Series
Explore reciprocals of A.P., telescoping series, and sum of squares/cubes of natural numbers.

Step 5: Recursive Patterns
Decode complex recurrence relations, invariant properties, and multi-layered functional equations.

Topic Hero: Sequence vs. Series

Sequence vs. Series

Sequence

An ordered list of numbers governed by a specific rule.

Notation: $a_{1}, a_{2}, \dots, a_{n}$
Focus: $n$ -th term ( $a_{n}$ )

Series

The sum of the terms of a sequence.

Notation: $S_{n} = \sum a_{i}$
Focus: Total ( $S_{n}$ )

Core Relationship:
The

n

-th term of any series can be extracted using the sum:

a_{n} = S_{n} - S_{n - 1} for n \geq 2

Arithmetic Progression: Core Properties

Arithmetic Progression

Definition:

a_{n} - a_{n - 1} = d

(constant common difference)

Standard Formulas

$n$ -th term: $a_{n} = a + (n - 1) d$
Sum of $n$ terms: $S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$

Exam-Level Properties

Symmetric Sum: $a_{k} + a_{n - k + 1} = a_{1} + a_{n}$
Arithmetic Mean: $2 a_{n} = a_{n - 1} + a_{n + 1}$
Linear Selection: Terms at regular intervals form an A.P.

Geometric Progression & Infinite Series

Geometric Progression

Definition:

\frac{a _{n}}{a _{n - 1}} = r

(constant common ratio)

Standard Formulas

$n$ -th term: $a_{n} = a \cdot r^{n - 1}$
Sum of $n$ terms: $S_{n} = {a \frac{r ^{n} - 1}{r - 1} a \frac{1 - r ^{n}}{1 - r} if r > 1 if r < 1$

Infinite Geometric Series: Converges ONLY if

∣ r ∣ < 1

S_{\infty} = \frac{a}{1 - r}

Geometric Mean:

a_{n}^{2} = a_{n - 1} \cdot a_{n + 1}

Sequences, Progressions and Patterns: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

In an arithmetic progression, the first term is 7 and the common difference is 4. What is the value of the 12th term?

A.
51
B.
48
C.
55
D.
44

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct application of the Arithmetic Progression (A.P.) $n$ -th term formula.

Step 1: Identify the given values. The first term $a = 7$ , the common difference $d = 4$ , and the term number $n = 12$ .

Step 2: Recall the formula for the $n$ -th term of an A.P.: $a_{n} = a + (n - 1) d$ .

Step 3: Substitute the values into the formula.

$a_{12} = 7 + (12 - 1) \times 4$

$a_{12} = 7 + 11 \times 4$

$a_{12} = 7 + 44 = 51$ .

Answer: 51

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

What is the sum of the first 10 terms of an arithmetic progression where the first term is 3 and the common difference is 2?

A.
120
B.
100
C.
130
D.
110

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall and application of the Arithmetic Progression (A.P.) sum formula.

Step 1: Identify the given values. The first term $a = 3$ , the common difference $d = 2$ , and the number of terms $n = 10$ .

Step 2: Recall the formula for the sum of the first $n$ terms of an A.P.: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Step 3: Substitute the values into the formula.

$S_{10} = \frac{10}{2} [2 (3) + (10 - 1) 2]$

$S_{10} = 5 [6 + 9 (2)]$

$S_{10} = 5 [6 + 18]$

$S_{10} = 5 [24] = 120$ .

Answer: 120

More notes in this unit

chapter

Sequences, Progressions and Patterns Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Sequences, Progressions and Patterns notes for XAT: 11 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

In an arithmetic progression, the first term is 7 and the common difference is 4. What is the value of the 12th term?

Question 2

What is the sum of the first 10 terms of an arithmetic progression where the first term is 3 and the common difference is 2?

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