Sequence of series practice · Medium

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

The sum of series $\frac{1}{{1.2.3}} + \frac{1}{{3.4.5}} + \frac{1}{{5.6.7}} + \ldots \infty$ is equal to

The sum to infinity of the following series $\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\ldots$. shall be-

The sum of all two digit numbers which, when divided by 4, yield unity as a remainder is

If the equation $(x+1)+(x+4)+(x+7)+\ldots+(x+34)=216$ represents the sum of the first $n$ terms of an A.P., then $x=$

The sum of all two digit odd numbers is

If $a, b, c$ are in A.P., then $2^{a x+1}, 2^{b x+1}, 2^{c x+1}, x \neq 0$ are in

The sum of the first $n$ terms of two arithmetic series are in the ratio $(7 n+1):(4 n+27)$. The ratio of their 11th terms is

$\sum\limits_{{i=1}}^n \sum\limits_{{j=1}}^i \sum\limits_{{k=1}}^j 1=$

If the sum to first $n$ terms of the AP 2, 4, 6,… is 240, then the value of $n$ is

A club consists of members whose ages are in AP, the common difference being 3 months. If the youngest member of the club is just 7 yr old and the sum of the ages of all the members is $250 \mathrm{yr}$, then the number of members in the club is

If ${a_1},{a_2}, \ldots ,{a_n}$ are in HP, then the expression ${a_1}{a_2} + {a_2}{a_3} + .. + {a_{n - 1}}{a_n}$ is equal to

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$ terms of a non-constant A.P. be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of a G.P. If the first term of the A.P. is 1 , then the sum of its first 20 terms is equal to

The sum of the series $\frac{3}{{4.8}} - \frac{{3.5}}{{4.8.12}} + \frac{{3.5.7}}{{4.8.12.16}} - \;...$

If $x=\sum_{n=0}^{\infty} a^{n}, y=\sum_{n=0}^{\infty} b^{n}, z=\sum_{n=0}^{\infty} c^{n}$ where $a, b, c$ are in

The sum of the first 10 terms of $1 \cdot x + 4 \cdot {x^2} + 7 \cdot {x^3} + .....$ is

If $a_1, a_2, \ldots . a_n$ are in H.P., then $\frac{a_1}{a_2+a_3+\ldots .+a_n}, \frac{a_2}{a_1+a_3+\ldots .+a_n}, \ldots \ldots \ldots \ldots, \frac{a_n}{a_1+a_2+\ldots .+a_{n-1}}$ are in

The sum of the series
$\frac{{x - 1}}{{x + 1}} + \frac{1}{2}\frac{{{x^2} - 1}}{{{{\left( {x + 1} \right)}^2}}} + \frac{1}{3}\frac{{{x^3} - 1}}{{{{\left( {x + 1} \right)}^3}}} + \ldots ,$ is equal to

If the arithmetic mean of two numbers $a$ and $b, a>b>0$, is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to :

Given that $\alpha, \gamma$ are roots of the equation $A x^2-4 x+1=0$ and $\beta, \delta$ are the roots of the equation $B x^2-6 x+1=0$, then values of $A$ and $B$ such that $\alpha, \beta, \gamma$ and $\delta$ are in HP, are

If $1, \log _y x, \log _z y$ and $-15 \log _x z$ are in $\mathrm{AP}$, then

The consecutive terms $\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}$ of a series are in

If $T_1, T_2, T_3, T_4, T_5, T_6, T_7, \ldots$ are in AP with common difference 2 . Then, the difference of any two consecutive terms of the sequence $T_1, T_4, T_7, \ldots$ is

If the AM of two numbers be $A$ and GM be $G$, then the numbers will be

$\sum \limits_{n = 1}^\infty \frac{1}{{\left( {n + 1} \right)\left( {n + 2} \right) \ldots \left( {n + k} \right)}}$ is equal to

The set of real values of $x$ satisfying $|x-1| \leq 3$ and $|x-1| \geq 1$ is

The sum of the series $1+2 \cdot 2+3 \cdot 2^2+4 \cdot 2^3+\ldots+100 \cdot 2^{99}$ is

Let a three term strictly increasing arithmetic sequence of integers are one such that the sum of the squares of the three terms equals the product of the middle term and the square of the common difference, then the sum of third term of such sequence is $p$, then $(p-25)$ is

The value of $\log 2 + 2\left( {\frac{1}{5} + \frac{1}{3} \cdot \frac{1}{{{5^3}}} + \frac{1}{5} \cdot \frac{1}{{{5^5}}} + \ldots + \infty } \right)$ is

The sum of $n$ terms of the series $\frac{5}{1.2} \cdot \frac{1}{3}+\frac{7}{2 \cdot 3} \cdot \frac{1}{3^2}+\frac{9}{3.4} \cdot \frac{1}{3^3}+\frac{11}{4.5} \cdot \frac{1}{3^4}+\ldots$. is: