Sequence of series practice · Mixed difficulty

If the sum
$\frac{3}{1^{2}}+\frac{5}{1^{2}+2^{2}}+\frac{7}{1^{2}+2^{2}+3^{2}}+\ldots \ldots+$ up to 20 terms is equal
to $\frac{\mathrm{k}}{21}$, then $\mathrm{k}$ is equal to:

If $a(b-c) x^2+b(c-a) x y+c(a-b) y^2$ is a perfect square, the quantities $a, b, c$ are in

If $a+b+c=3$ and $a > 0, b > 0, c > 0$, then the greatest value of $a^2 b^3 c^2$ is

If first three terms of sequence $\frac{1}{16}, a, b, \frac{1}{6}$ are in geometric series and the last three terms are in harmonic series, then the values of $a$ and $b$ will be

For $x \in \mathbf{R}$, let $[x]$ denote the greatest integer $\leq x$. Largest natural number $n$ for which

$E=\left[\frac{\pi}{2}\right]+\left[\frac{1}{100}+\frac{\pi}{2}\right]+\left[\frac{2}{100}+\frac{\pi}{2}\right] \ldots+\left[\frac{n}{100}+\frac{\pi}{2}\right]<43,$ is

If $m$ is the AM of two distinct real numbers $l$ and $n$ $(l, n>1)$ and $G_1, G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2 G_2^4+G_3^4$ equals

$1+\frac{\log _{e} x}{1 !}+\frac{\left(\log _{e} x\right)^{2}}{2 !}+\frac{\left(\log _{e} x\right)^{3}}{3 !}+\cdots \infty=$

If twice the 11th term of an AP is equal to 7 times its 21st term, then its 25th term is equal to

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 the sum of $p$ terms of an AP is $q$ and that of $q$ terms is $p$, then the sum of $p+q$ terms will be

The value of $0.4 \overline{23}$ is

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

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-6 x^2+11 x-6=0$. Then, HM of the roots is

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 :

The sum of $n$ terms of the series $1+(1+3)+(1+3+5)+\ldots \ldots$, is

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

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

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

If ${a_1},{a_2},{a_3}, \ldots ,\;{a_n}$ are in AP, where ${a_i} > 0$ for all i, then value of $\frac{1}{{\sqrt {{a_1}} + \sqrt {{a_2}} }} + \frac{1}{{\sqrt {{a_2}} + \sqrt {{a_3}} }} + ... + \frac{1}{{{a_{n - 1}} + \sqrt {{a_n}} }}$ is equal to

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:

${1^3} - {2^3} + {3^3} - {4^3} + \ldots + {9^3} =$

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

Jairam purchased a house in Rs 15000 and paid Rs 5000 at once. Rest money he promised to pay in annual installment of Rs 1000 with 10% per annum interest. How much money is to be paid by Jairam?

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