List of top Mathematics Questions on Sequence and series

If the sum of the first $ n $ terms of an arithmetic progression (AP) is given by $ S_n = 3n^2 + 2n $, find the 4th term of the AP.

If the sum of the first 20 terms of the series $$ \frac{4.1}{4 + 3.1^2 + 1^4} + \frac{4.2}{4 + 3.2^2 + 2^4} + \frac{4.3}{4 + 3.3^2 + 3^4} + \frac{4.4}{4 + 3.4^2 + 4^4} + ... $$ is $ \frac{m}{n} $, where $ m $ and $ n $ are coprime, then $ m + n $ is equal to:

The sum of the infinite series $ \cot^{-1} \left( \frac{7}{4} \right) + \cot^{-1} \left( \frac{19}{4} \right) + \cot^{-1} \left( \frac{39}{4} \right) + \cot^{-1} \left( \frac{67}{4} \right) + ... $ is:

For positive integers $ n $, if $ 4 a_n = \frac{n^2 + 5n + 6}{4} $ and \[ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} \]

If the sum of the first 10 terms of the series \[ \frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots \] is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to _____.

If $ \alpha $ is a root of the equation $ x^2 + x + 1 = 0 $ and $$ \sum_{k=1}^{n} \left( \alpha^k + \frac{1}{\alpha^k} \right)^2 = 20$$, then n is equal to _______

If $\sum$$_{r=1}^n T_r$ = $\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$ , then $ \lim_{n \to \infty} \sum_{r=1}^n \frac{1}{T_r} $ is equal to :

The sum of the infinite geometric series $ S = \frac{a}{1-r} $ is 24, and the sum of the first three terms is 21. Find $ a $ and $ r $.

If $ \alpha, \beta $ are the roots of the equation $ x^2 - x - 1 = 0 $ and $ S_n = 2023 \alpha^n + 2024 \beta^n $, then:

Let $ \alpha, \beta $ be the distinct roots of the equation $$ x^2 - (t^2 - 5t + 6)x + 1 = 0, \, t \in \mathbb{R} \, \text{and} \, a_n = \alpha^n + \beta^n. $$ Then the minimum value of $ \frac{a_{2023} + a_{2025}}{a_{2024}} $ is:

Let 2nd, 8th, and 44th terms of a non-constant A.P. be respectively the 1st, 2nd, and 3rd terms of a G.P. If the first term of A.P. is 1, then the sum of the first 20 terms is equal to

The number of common terms in the progressions 4, 9, 14, 19, ...,up to 25th term and 3, 6, 9, 12, ..., up to 37th term is:

The function $ f(x) = 2x + 3(x)^{\frac{2}{3}}, x \in \mathbb{R} $, has

Given an A.P. and a G.P. with positive terms, with the first and second terms of the progressions being equal. If $a_n$ and $b_n$ be the $n$-th term of A.P. and G.P. respectively, then:

If for the series $a_1, a_2, a_3, \ldots$, etc., $a_{n+1} - a_n$ bears a constant ratio with $a_n + a_{n+1}$, then $a_1, a_2, a_3, \ldots$ are in:

If $ a_1, a_2, \dots, a_n $ are in A.P. with common difference $ \theta $, then the sum of the series:
\[ \sec a_1 \sec a_2 + \sec a_2 \sec a_3 + \dots + \sec a_{n-1} \sec a_n = k (\tan a_n - \tan a_1), \]
where $ k = ? $

If three successive terms of a G.P. with common ratio $r \, (r>1)$ are the lengths of the sides of a triangle and $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$, then $3\lfloor r \rfloor + \lfloor -r \rfloor$ is equal to:

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$, then $S_{15} - S_{5}$ is equal to:

Let $ x_1, x_2, x_3, x_4 $ be the solution of the equation \[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 \] and \[ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m.\] Then the value of $ m $ is ______.

If $\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}} = m$ and $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n,$ then the point $ (m, n) $ lies on the line

The sum of the series \[ \frac{1}{1 - 3 \cdot 1^2 + 1^4} + \frac{2}{1 - 3 \cdot 2^2 + 2^4} + \frac{3}{1 - 3 \cdot 3^2 + 3^4} + \ldots \text{ up to 10 terms} \] is

Let $ S_n $ be the sum to $ n $-terms of an arithmetic progression $ 3, 7, 11, \ldots $.
If $ 40 < \left( \frac{6}{n(n+1)} \sum_{k=1}^{n} S_k \right) <42 $, then $ n $ equals ___.

$3, a, b, c$ are in Ap and $3, a-1, b+1, c+9$ are in GP. Then AM of $a, b, c$ is

How many times 3 comes from 1 to 1000?

Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in$ N If $f(1)=3$ and $\displaystyle\sum_{k=1}^n f(k)=3279$, then the value of $n$ is