Sequence of series practice · Easy

The sum of the series $1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+2.6^{2}+\ldots .+2(2 m)^{2}$ is

If one A.M. ' $a$ ' and two G.M.'s $p$ and $q$ be inserted between any two numbers, then the value of $p^3+q^3$ is

If $\sum\limits_{{k=1}}^n\left(\sum\limits_{{m=1}}^k m^2\right)=a n^4+b n^3+c n^2+d n+e$, then

If $\frac{{\log a}}{{b - c}} = \frac{{\log b}}{{c - a}} = \frac{{\log c}}{{a - b}},$ then ${a^a}{b^b}{c^c} =$

If $a x^3+b x^2+c x+d$ is divisible by $a x^2+c$, then $a, b, c, d$ are in

The nth and $10 t h$ terms of $a G P$ $5,25,125, \ldots$, are respectively

Let ${A_1},{A_2},{A_3}, \ldots$ be an increasing geometric progression of positive real numbers. If ${A_1}{A_3}{A_5}{A_7} = \frac{1}{{1296}}$ and ${A_2} + {A_4} = \frac{7}{{36}},$ then, the value of ${A_6} + {A_8} + {A_{10}}$ is equal to

If sum of the series $\sum \limits_{n = 0}^\infty {r^n} = S$ for $\left| r \right| < 1$, then sum of the series $\sum \limits_{n = 0}^\infty {r^{2n}}$ , is

Let $\left\{a_n\right\}_{n=0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=2 a_{n+1}-a_n+1$ for all $n \geq 0$.
Then $\sum\limits_{n=2}^{\infty} \frac{a_n}{7^n}$ is equal to :

If three unequal numbers $p,\;q,\;r$ are in HP and their squares are in AP, then the ratio $p\;:q\;:r$ is

The sum of $n$ terms of an A.P. is $3{n^2} + 5.$ The number of term which equals 159 is

For three positive integers $p, q, r, x^{p q^{2}}=y^{q r}=z^{p^{2} r}$ and $r=p q+1$ such that $3,3 \log _{y} x, 3 \log _{z} y, 7 \log _{x}$ $z$ are in A.P with common difference $\frac{1}{2}$. Then $r-p-$ $q$ is equal to

If ${a_1},{a_2},{a_3}, \ldots ,{a_n}$ are the $n$ arithmetic means between $a$ and $b$, then $2\sum \limits_{i = 1}^n {a_i}$ equals

Let a, b, c, d and p be any non-zero distinct real numbers such that $({a^2} + {b^2} + {c^2}){p^2} - 2(ab + bc + cd)p + ({b^2} + {c^2} + {d^2}) = 0.$ Then,

The value of ${\log _2}{\log _2}{\log _4}256 + 2{\log _{\sqrt 2 }}2,$ is

For $x \in R,\,\mathrm{let}\left[ x \right]$ denote the greatest integer $\le x$, then the sum of the series
$\left[ { - \frac{1}{3}} \right] + \left[ { - \frac{1}{3} - \frac{1}{{100}}} \right] + \left[ { - \frac{1}{3} - \frac{2}{{100}}} \right] + ... + \left[ { - \frac{1}{3} - \frac{{99}}{{100}}} \right]$ is

If ${\log _4}2 + {\log _4}4 + {\log _4}16 + {\log _4}x = 6,$ then $x =$

Suppose for each $n \in \mathbf{N}, 1^4+2^4+3^4+\ldots+n^4=a n^5+b n^4+c n^3+d n^2+e n+f$, then value of $b$ is

The number of solutions of ${\log _2}\left( {x - 1} \right) = 2{\log _2}\left( {x - 3} \right)$ is

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \ldots, \mathrm{a}_{30}$ be an A.P., $\mathrm{S}=\sum_{i=1}^{30} \mathrm{a}_{i}$ and $\mathrm{T}=\sum_{i=1}^{15} \mathrm{a}_{(2 i-1)}$
If $\mathrm{a}_{5}=27$ and $\mathrm{S}-2 \mathrm{T}=75$, then $\mathrm{a}_{10}$ is equal to:

If $2 x, x+8,3 x+1$ are in AP, then the value of $x$ will be

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30 . If the last term exceeds the first term by $10 \frac{1}{2}$, then the number of terms in the A.P. is:

If the sum of the first $2 n$ terms of $2,5,8 \ldots$ is equal to the sum of the first $n$ terms of $57,59,61 \ldots$. , then $n$ is equal to

If $x,y,z$ are in AP, then $\frac{1}{{\sqrt x + \sqrt y }},\frac{1}{{\sqrt z + \sqrt x }},\frac{1}{{\sqrt y + \sqrt z }}$ are in

If $a,\;b,\;c,\;d,\;e,\;f\;$ are in AP, then the value of $e - c$ will be

The number of roots of the equation $x^{\log _x(x+3)^2}=16$ is

$0.5737373 \cdots=$

If $\log _{0.3}(x-1)<\log _{0.09}(x-1),$ then $x$ lies in the interval

If the sum of an infinitely decreasing G.P. is 3, and the sum of the squares of its terms is $9/2,$ the sum of the cubes of the terms is

If $a, b, c, d, e, f$ are in $\mathrm{AP}$, then the value of $e-c$ will be