Sequence of series practice · Mixed difficulty

It is given that $\frac{1}{{{1^4}}} + \frac{1}{{{2^4}}} + \frac{1}{{{3^4}}} + ... + \mathrm{to}\;\infty = \frac{{{\pi ^4}}}{{90}}$. Then, $\frac{1}{{{1^4}}} + \frac{1}{{{3^4}}} + \frac{1}{{{5^4}}} + ...\infty$ is equal to

Insert two harmonic means between 3 and 5 .

The value of ${\log _5}\left( {1 + \frac{1}{5}} \right) + {\log _5}\left( {1 + \frac{1}{6}} \right) + {\log _5}\left( {1 + \frac{1}{7}} \right) + \ldots$

$+ {\log _5}\left( {1 + \frac{1}{{624}}} \right)\;\mathrm{is}$

If $a,b,c$ are in G.P., then ${\log _a}x,{\log _b}x,{\log _c}x$ are in

The coefficient of ${x^n}$ in the series

$1 + \frac{{a + bx}}{{1!}} + \frac{{{{\left( {a + bx} \right)}^2}}}{{2!}} + \frac{{{{\left( {a + bx} \right)}^3}}}{{3!}} + \ldots ,$ is

If $H_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, then value of $1+\frac{3}{2}+\frac{5}{3}+\ldots+\frac{2 n-1}{n}$ is

The sum $\sum_{k=1}^{20} k \frac{1}{2^{k}}$ is equal to :

The sum to $n$ terms of the series $\frac{1}{\sqrt{7}+\sqrt{10}}+\frac{1}{\sqrt{10}+\sqrt{13}}+\frac{1}{\sqrt{13}+\sqrt{16}}+\cdots$ is

The value of $\sum \limits_{r = 1}^n \log \left( {\frac{{{a^r}}}{{{b^{r - 1}}}}} \right)$ is

Let two numbers have arithmetic mean 9 and geometric mean 4. Then, these numbers are the roots of the quadratic equation

If $a, b$ and $c$ are in HP, then the straight line $\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0$ always passes through a fixed point and that point is

Let $f: R \rightarrow R$ be a function which satisfies $f(x+y)=f(x)+f(y)$ for all $x, y \in R$. If $f(1)=2$ and $g(n)=\sum\limits_{k=1}^{n-1} f(k), n \in N$, then the value of $n$ for which $g(n)=20$, is

The consecutive numbers of a three digit number form a GP. If we subtract 792 from this number, then we get a number consisting of the same digits written in the reverse order and if we increase the second digit of the required number by 2 , then resulting number forms an AP. The number is

The sum of 10 terms of the series $\frac{3}{{{1^2} \times {2^2}}} + \,\frac{5}{{{2^2} \times {3^2}}} + \,\frac{7}{{{3^2} \times {4^2}}} + ........$ is

If $f(x)$ is a function satisfying $f(x+y)=f(x) f(y), \forall x, y \in N$ such that $f(1)=3$ and $\sum_{x=1}^n f(x)=120$. Then, the value of $n$ is

If $S_n=\sum_{r=0}^n \frac{1}{{ }^n C_r}$ and $t_n=\sum_{r=0}^v \frac{r}{{ }^r C_r}$, then $\frac{t_n}{S_s}$ is equal to:

If $n$ is odd, the sum of $n$ terms of the series $1^2-2^2+3^2-4^2+5^2-6^2+\ldots \ldots$, is

If $a, b, c$ are in A.P. and $a^{2}, b^{2}, c^{2}$ are in G.P. such that $a < b < c$ and $a+b+c=\frac{3}{4}$, then the value of $a$ is

If ${S_n} = \sum \limits_{r = 1}^n {a_r} = \frac{1}{6}n\left( {2{n^2} + 9n + 13} \right),$ then $\sum \limits_{r = 1}^n \sqrt {{a_r}}$ equals

If $x = 1 + a + {a^2} + \ldots \infty$ and $y = 1 + b + {b^2} + \ldots \infty$ where $a$ and $b$ are proper fractions, then $1 + ab + {a^2}{b^2} + \ldots \infty$ equals

If $y = {2^{1/{{\log }_x}8}},$, then $x$ equal to

If the sum of first $n$ terms of an AP is $c n^2$, then the sum of squares of these $n$ terms is

The minimum value of ${4^x} + {4^{1 - x}},x \in R,$ is

If $S_n$ denotes the sum of $n$ terms of an AP, then $S_{n+3}-3 S_{n+2}+3 S_{n+1}-S_n$ is equal to

If $n$ arithmetic means are inserted between $a$ and 100 such that the ratio of the first mean to the last mean is 1: 7 and $a + n = 33,$ then the value of $n$ is:

$\frac{1}{2} - \frac{1}{{{{2.2}^2}}} + \frac{1}{{{{3.2}^3}}} - \frac{1}{{{{4.2}^4}}} + \ldots$ is equal to

The sum of the series $04+0.004+000004+\ldots$ is

The value of $n$ for which $704+\frac{1}{2}(704)+$ $\frac{1}{4}(704)+\ldots$ upto $n$ terms $=1984-\frac{1}{2}(1984)+\frac{1}{4}(1984)$ $-\ldots$ upto $n$ terms is

Which of the following is the sum of 20 terms of an $A P 1,4,7,10, \ldots$ ?

If $f(n)=\left[\frac{1}{2}+\frac{n}{100}\right]$, where $[x]$ denotes the integral part of $x$, then the value of $\sum\limits_{{n=1}}^{100} f(n)$ is