List of top Quantitative Aptitude Questions on Sequences and Series asked in CAT

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) $?

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:

Which of the following best describes $ a_n + b_n $ for even $ n $?

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 $?

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:

If $a_1=1$ and $a_{n+1} - 3a_n + 2 = 4n$, find $a_{100}$.

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?

There are two disjoint sets $S_1$ and $S_2$: $S_1 = \{ f(1), f(2), f(3), \dots \}$, $S_2 = \{ g(1), g(2), g(3), \dots \}$ such that $S_1 \cup S_2$ = natural numbers. Also $f(1)<f(2)<f(3)<\dots$ and $g(1)<g(2)<g(3)<\dots$, and $f(n) = g(g(n)) + 1$. Find $g(1)$.

Evaluate:
$\frac{1}{2^2 - 1} + \frac{1}{4^2 - 1} + \frac{1}{6^2 - 1} + \dots + \frac{1}{20^2 - 1}$

Each of the numbers $x_1, x_2, \dots, x_n, \ n \ge 4$, is equal to $1$ or $-1$. Suppose, \[ x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 + x_3 x_4 x_5 x_6 + \dots + x_{n-3} x_{n-2} x_{n-1} x_n + x_{n-2} x_{n-1} x_n x_1 + x_{n-1} x_n x_1 x_2 + x_n x_1 x_2 x_3 = 0, \] then which of the following is true?

For real number $ x $, let

\[ f(x) = \begin{cases} \frac{1}{1+x}, & \text{if } x \text{ is non-negative} \\ 1 + x, & \text{if } x \text{ is negative} \end{cases} \]

\[ f^n(x) = f\left(f^{n-1}(x)\right), \quad n = 2, 3, \dots \]

A series $S_{1}$ of five positive integers is such that the third term is half the first term and the fifth term is $20$ more than the first term. In series $S_{2}$, the $n$th term defined as the difference between the $(n+1)$th term and the $n$th term of series $S_{1}$, is an arithmetic progression with a common difference of $30$.