List of top Mathematics Questions on Sequence and series

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?

If \(a_1,a_2,.......a_n\) are in arithmetic progression with common difference \(d>n\), then find limit \(\lim_{n \to \infty}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{a_1}+\sqrt{a_2}})+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+.......+\frac{1}{\sqrt{a_n}+\sqrt{a_{n-1}}})\) is____.

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

The sum of the first 20 terms of the series 5 + 11 + 19 + 29 + 41 + ... is

\(\frac{nC_n}{(n+1)}\) + \(\frac{nC_{n-1}}{(n)}\) + ……… + ½ \(nC_1\) + \(nC_0\) = \(\frac{255}{8}\), Then value of n is

Let be a sequence such that a₁ + a₂ +....+ a_n=\(\frac{n^2+3n}{(n+1) (n+2)}\) If  \(28 ∑^{10}_{k=1} \frac{1}{a_k} =  P _1 P_ 2 P _3 . . . P_ m\) , where p₁, p₂, …..p_m are the first m prime numbers, then m is equal to

Consider a function f:N→R, satisfying
$ f(1)+2f(2)+3f(3)+⋯+xf(x)=x(x+1)f(x);x≥2 with f(1)=1.$
Then $\frac{1}{f(2022)} + \frac{1}{f(2028)}$ is equal to $

If \(a_n=\frac{-2}{4n^2-16n+15}\) and a₁ + a₂ + …… + a₂₅ = m/n where m and n are coprime, then the value of m + n is

Let \(𝑎_1=𝑏_1=1\)and \(𝑎_𝑛=𝑎_{𝑛−1}+(𝑛−1)\), \(𝑏_𝑛=𝑏_{𝑛−1}+𝑎_{𝑛−1}\), \(∀𝑛≥2\). If \(𝑆= \displaystyle\sum_{ 𝑛=1}^{10}\frac {𝑏_𝑛}{2^𝑛}\) and \(𝑇= \displaystyle\sum_{𝑛=1}^{8} \frac {𝑛}{2^{𝑛−1}}\), then \(2^7(2𝑆−𝑇)\)  is equal to ____ .

Let \({a_k}\) and \({b_k}\), \(𝑘∈N\), be two G.P.s with common ratios \(r_1\) and \(r_2\) respectively such that \(a_1= b_1=4\) and \(r_1<r_2\). Let \(c_k=a_k+b_k\), \(k∈N\). If \(c_2=5\) and \(c_3=\frac {13}{4}\), then \( \displaystyle\sum_{k=1}^{∞ }c_k−(12a_6+8b_4)\) is equal to _____ .

For \(k ∈ N\), if the sum of the series \(1 + \frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+..\)..  is 10, then the value of k is___

n^th term of the series
\(1+\frac{3}{7}+\frac{5}{7^2}+\frac{1}{7^2}+...\) is

The value of \(\lim_{n\rightarrow\infty}[(\frac{1}{2.3}+\frac{1}{2^2.3})+(\frac{1}{2^2.3^2}+\frac{1}{2^3.3^2})+.....+(\frac{1}{2^n.3^n}+\frac{1}{2^{n+1}.3^n})]\)