List of top Mathematics Questions on Sequences and Series

Let the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}\), where \(a>2\). If \(a,4,b\) are in A.P., then the equation \[ ax^2-ax+2(a-2b)=0 \] has:

In a G.P., if the product of the first three terms is \(27\) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a,b) \), then \( a^2+b^2 \) is equal to:

The value of \[ \sum_{k=1}^{\infty} (-1)^{k+1}\left(\frac{k(k+1)}{k!}\right) \] is:

Let \(729, 81, 9, 1, \ldots\) be a sequence and \(P_n\) denote the product of the first \(n\) terms of this sequence. If \[ 2 \sum_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^{\alpha} - 1}{3^{\beta}} \] and \(\gcd(\alpha, \beta) = 1\), then \(\alpha + \beta\) is equal to

Let \[ \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n. \] If \( a_{10} = 59 \) and \( a_6 = 7a_1 \), then \( \alpha + \beta \) is equal to

Let the area of the region bounded by the curve \( y = \max \{ \sin x, \cos x \} \), lines \( x = 0 \), \( x = 3\pi/2 \), and the x-axis be A. Then, \( A + A^2 \) is equal to :

Among the statements :
I: If the given determinants are equal, then \( \cos^2\alpha + \cos^2\beta + \cos^2\gamma = \frac{3}{2} \), and
II: If the polynomial determinant equals \( px + q \), then \( p^2 = 196q^2 \), identify the truth value.

Suppose \(a, b, c\) are in A.P. and \(a^2, 2b^2, c^2\) are in G.P. If \(a<b<c\) and \(a+b+c = 1\), then \(9(a^2+b^2+c^2)\) is equal to ____________.

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \dots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1 + a_2 + \dots + a_{10} = 62$, then $a_1$ is equal to :

The positive integer \( n \), for which the solutions of the equation \( x(x+2) + (x+2)(x+4) + \dots + (x+2n-2)(x+2n) = \frac{8n}{3} \) are two consecutive even integers, is :

Let \( a_1 = 1 \) and for \( n \ge 1 \), \[ a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2}. \] Then \[ \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \] is equal to

Let \(a_1, a_2, a_3, \dots\) be a G.P. of increasing positive terms such that \(a_2 \cdot a_3 \cdot a_4=64\) and \(a_1 + a_3 + a_5 = \frac{813}{7}\). Then \(a_3 + a_5 + a_7\) is equal to:

If \( \sum_{k=1}^{n} a_k = \alpha n^2 + \beta n \) and \( a_{10} = 59,\; a_6 = 7a_1 \), then find \( \alpha + \beta \):

If sum of first 4 terms of an A.P. is 6 and sum of first 6 terms is 4, then sum of first 12 terms of an A.P. is

If \(a_1 = 1\) and for all \(n \ge 1\), \[ a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2}, \] then the value of \[ \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \] is equal to:

Let \( f(x) = \lim_{n \to \infty} \left( \frac{1}{n^3} \sum_{k=1}^{n} \left\lfloor \frac{k^2}{3^x} \right\rfloor \right) \), where \( \left\lfloor . \right\rfloor \) denotes the greatest integer function, then \( 12 \sum_{j=1}^{\infty} f(j) \) is equal to:

If \( a_1 = 1 \) and for \( n \geq 1 \), \[ a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \] then \[ \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \] is equal to

If \( a_1, a_2, a_3, \dots \) are the terms of an increasing geometric progression such that \[ a_1 + a_3 + a_5 = 21, \quad a_1a_3a_5 = 64 \] then \[ a_1 + a_2 + a_3 \] is

If the sum of first 4 terms of an AP is 6 and sum of first 6 terms is 4, then sum of first 12 terms of AP is :

1 + 3 + $5^2$ + 7 + $9^2$ + $\ldots$ upto 40 terms is equal to

For any positive integer \( n \), the value of \( 1 + 1! + 2.2! + 3.3! + \cdots + n.n! \) will be