Sequences, Progressions and Patterns Practice Questions for XAT: 86+ Solved Questions with Step-by-Step Solutions

Solve 86+ Sequences, Progressions and Patterns practice questions for XAT with answers and detailed solutions. Free sample questions below.

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

Sequences, Progressions and Patterns: Solved Questions with Step-by-Step Explanations (5 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

Question 3 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

Which formula correctly gives the $n$ -th term of a geometric progression?

A.
$a_{n} = a + (n - 1) d$
B.
$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$
C.
$S_{n} = \frac{n}{2} (a + l)$
D.
$a_{n} = a \cdot r^{n - 1}$

Correct Answer:

Step-by-Step Solution

Key idea: This is a quick formula recall question for the $n$ -th term of a Geometric Progression (G.P.).

Step 1: Recall the definition of a G.P. Each term is obtained by multiplying the previous term by a constant common ratio $r$ .

Step 2: The first term is $a$ . The second term is $a r$ . The third term is $a r^{2}$ .

Step 3: Generalize this pattern to the $n$ -th term: $a_{n} = a \cdot r^{n - 1}$ .

Step 4: Match this with the given options. Option D is the correct formula.

Answer: $a_{n} = a \cdot r^{n - 1}$

Question 4 · Quantitative Aptitude and Data Interpretation (QA & DI) NAT

Evaluate the sum:

$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{99 \cdot 100}$

Correct Answer:

0.99

Step-by-Step Solution

Key idea: This is a telescoping series. Each term can be split into partial fractions.

Step 1: Decompose the general term.

$\frac{1}{n ( n + 1 )} = \frac{1}{n} - \frac{1}{n + 1}$ .

Step 2: Write out the sum.

$S = (\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + \dots + (\frac{1}{99} - \frac{1}{100})$ .

Step 3: Cancel intermediate terms.

The $- \frac{1}{2}$ cancels with $+ \frac{1}{2}$ , $- \frac{1}{3}$ with $+ \frac{1}{3}$ , and so on.

Only the first part of the first term and the second part of the last term remain.

Step 4: Calculate the result.

$S = 1 - \frac{1}{100} = \frac{99}{100} = 0.99$ .

Question 5 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

According to the exam heuristics for this chapter, what is the recommended first algebraic step when you encounter a recursive sequence like $a_{n + 1} = \frac{1}{1 + a _{n}}$ ?

A.
Assume it is a geometric progression
B.
Plot the first 10 terms on a graph
C.
Differentiate the relation with respect to $n$
D.
Invert or shift the relation to find a telescoping pattern

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question testing the specific exam heuristics provided in the chapter summary.

Step 1: Identify the pattern.

The sequence $a_{n + 1} = \frac{1}{1 + a _{n}}$ is a fractional recursive relation.

Step 2: Recall the heuristic.

The summary card explicitly states: "If you see a recursive sequence, invert or shift the relation to find a telescoping pattern."

Step 3: Apply to the specific example.

Inverting gives $\frac{1}{a _{n + 1}} = 1 + a_{n}$ , which is easier to manipulate.

Answer: D

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?

Question 3

Which formula correctly gives the n -th term of a geometric progression?

Question 4

Evaluate the sum: \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{99 \cdot 100}

Question 5

According to the exam heuristics for this chapter, what is the recommended first algebraic step when you encounter a recursive sequence like a_{n + 1} = \frac{1}{1 + a _{n}} ?

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Sequences, Progressions and Patterns Practice Questions for XAT: 86+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Sequences, Progressions and Patterns

Chapter Roadmap

Topic Hero: Sequence vs. Series

Sequence vs. Series

Sequence

Series

Sequences, Progressions and Patterns: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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