The CAT QA section requires speed and accuracy, along with a thorough understanding of the Sequences and Series. This article provides a set of MCQs on Sequences and Series to help you understand the topic and improve your problem-solving skills with the help of detailed solutions by ensuring conceptual clarity, which will help you in the CAT 2025 exam preparation
Whether you're revising the basics or testing your knowledge, these MCQs will serve as a valuable practice resource.
The CAT 2025 exam is expected to follow a similar trend to the CAT 2024, with 22 questions from the QA section out of a total of 68 questions.
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CAT MCQs on Sequences and Series
1. Let \( f_{n+1}(x) = f_n(x) + 1 \) if \( n \) is a multiple of 3; otherwise, \( f_{n+1}(x) = f_n(x) - 1 \).
If \( f_1(1) = 0 \), then what is \( f_{50}(1) \)?
If \( f_1(1) = 0 \), then what is \( f_{50}(1) \)?
A
-18
B
-16
C
-17
D
Cannot be determined
2. Let \(S_n\) denote the sum of the squares of the first \(n\) odd natural numbers. If \(S_n = 533n\), find the value of \(n\).
A
18
B
20
C
24
D
30
3. Let \( S \) denote the infinite sum:\[S = 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \ldots \quad \text{where } |x|<1\]and the coefficient of \( x^n \) is \( \frac{1}{2}n(n+3) \). Then \( S \) equals:
A
\( \frac{2 - x}{(1 + x)^3} \)
B
\( \frac{2 - x}{(1 - x)^3} \)
C
\( \frac{2x}{(1 - x)^3} \)
D
\( \frac{2 + x}{(1 + x)^3} \)
4. The sum of the first n terms of an arithmetic progression is \(3n^2 + 2n\). What is the 10th term?
A
59
B
60
C
61
D
62
5. A sum of money doubles in 5 years at simple interest. In how many years will it become 4 times?
A
10 years
B
12 years
C
15 years
D
20 years
6. A sum of Rs. 2000 is invested at 10% per annum compound interest. What is the amount after 2 years?
A
Rs. 2400
B
Rs. 2420
C
Rs. 2440
D
Rs. 2460
7. If \(a_1=1\) and \(a_{n+1} - 3a_n + 2 = 4n\), find \(a_{100} \).
A
\(3^{99} - 200\)
B
\(3^{99} + 200\)
C
\(3^{100} - 200\)
D
\(3^{100} + 200\)
8. Consider a sequence where the $n$th term \(t_n = \frac{n}{n+2}$, $n = 1, 2, \dots\)
The value of \(t_3 \times t_4 \times t_5 \times \dots \times t_{53}\) equals:
The value of \(t_3 \times t_4 \times t_5 \times \dots \times t_{53}\) equals:
A
\(\frac{2}{495}\)
B
\(\frac{2}{477}\)
C
\(\frac{12}{55}\)
D
\(\frac{1}{1485}\)
9. Which of the following best describes \( a_n + b_n \) for even \( n \)?
A
\( q (pq)^{\frac{n}{2} - 1} (p + q) \)
B
\( qp^{\frac{n}{2} - 1}(p + q) \)
C
\( q^{\frac{n}{2}}(p + q) \)
D
\( q^{\frac{n}{2}}(p + q)^{\frac{n}{2}} \)
10. If \( p = \frac{1}{3} \) and \( q = \frac{2}{3} \), then what is the smallest odd \( n \) such that \( a_n + b_n<0.01 \)?
A
7
B
13
C
11
D
9
11. For a Fibonacci sequence, from the third term onwards, each term is the sum of the previous two. If the difference in squares of the 7th and 6th terms is 517, what is the 10th term?
A
147
B
76
C
123
D
Cannot be determined
12. Evaluate:
\(\frac{1}{2^2 - 1} + \frac{1}{4^2 - 1} + \frac{1}{6^2 - 1} + \dots + \frac{1}{20^2 - 1}\)
\(\frac{1}{2^2 - 1} + \frac{1}{4^2 - 1} + \frac{1}{6^2 - 1} + \dots + \frac{1}{20^2 - 1}\)
A
9/19
B
10/19
C
10/21
D
11/21
13. There are two disjoint sets \( S_1 \) and \( S_2 \); \( S_1 = \{ f(1), f(2), f(3), \ldots \} \), \( S_2 = \{ g(1), g(2), g(3), \ldots \} \) such that \( S_1 \cup S_2 = \mathbb{N} \). Also \( f(1) < f(2) < f(3) < \cdots \) and \( g(1) < g(2) < g(3) < \cdots \), and \( f(n) = g(g(n)) + 1 \). Find \( g(1) \).
A
0
B
1
C
2
D
Can't be determined
14. Each of the numbers \( x_1, x_2, \ldots, x_n \), with \( n \ge 4 \), is equal to \( 1 \) or \( -1 \). Suppose: \[ \begin{aligned} x_1x_2x_3x_4 &+ x_2x_3x_4x_5 \\ &+ x_3x_4x_5x_6 + \cdots \\ &+ x_{n-3}x_{n-2}x_{n-1}x_n \\ &+ x_{n-2}x_{n-1}x_nx_1 \\ &+ x_{n-1}x_nx_1x_2 \\ &+ x_nx_1x_2x_3 = 0 \end{aligned} \] then which of the following is true?
A
n is even
B
n is odd
C
n is an odd multiple of 3
D
n is prime
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