Sequence of series practice · Easy

If $\frac{1}{{{{\log }_x}10}} = \frac{2}{{{{\log }_a}10}} - 2,$ then $x =$

If $S_{n}=\sum_{r=0}^{n} \frac{1}{{ }^{n} C_{r}}$ and $t_{n}=\sum_{r=0}^{n} \frac{r}{{ }^{n} C_{r}}$, then $\frac{t_{n}}{S_{n}}$ is equal to

The expansion of $\log \left( {1 + 3x + 2{x^2}} \right)$ is

If $\frac{{3 + 5 + 7 + \ldots + n\;\mathrm{terms}}}{{5 + 8 + 11 + \ldots + 10\;\mathrm{terms}}} = 7,$ then the value of $n$ is

If ${S_n} = nP + \frac{1}{2}n\left( {n - 1} \right){\cal Q}$, where ${S_n}$ denotes the sum of the first $n$ terms of an AP, then the common difference is

If the sum of the series $2+\frac{5}{x}+\frac{25}{x^2}+\frac{125}{x^3}$ $+\ldots$ is finite, then

The sum of series $1 + \frac{1}{{4.2!}} + \frac{1}{{16.4!}} + \frac{1}{{64.6!}} + \ldots \infty$ is

The sum of all the numbers of the form $n^3$ which lie between 100 and 10000 , is

If ${\log _{10}}x = y,$ then ${\log _{{{10}^3}}}{x^2}$ equals

Let $a_1, a_2, \ldots$ be in AP and $q_1, q_2, \ldots$ be in GP. If $a_1=q_1=2$ and $a_{10}=q_{10}=3$, then

Sum of the series $S=1+\frac{1}{2}(1+2)+$ $\frac{1}{3}(1+2+3)+\frac{1}{4}(1+2+3+4)+\ldots$ upto 20 terms is

The sum to $n$ terms of the series $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\ldots$ is

If $\frac{{a + b}}{2},b,\frac{{b + c}}{2}$ are in HP, then $a,b,c$ are in

If $a>0$, then $\sum\limits_{n=1}^{\infty}\left(\frac{a}{a+1}\right)^n$ equals

If $a_1, a_2, \ldots$ are in HP and $f_k=\sum\limits_{r=1}^n a_r-a_k$, then $2^{\alpha_1}, 2^{\alpha_2}, 2^{\alpha_3}, 2^{\alpha_4}, \ldots$ are in
$\left\{ {{\text{Where, }}\,{\text{ }}{\alpha _1} = \frac{{{a_1}}}{{{f_1}}},\,\,\,{\alpha _2} = \frac{{{a_2}}}{{{f_2}}},\,\,\,{\alpha _3} = \frac{{{a_3}}}{{{f_3}}}, \ldots } \right\}$

Let $a_1=0$ and $a_1, a_2, a_3, \ldots, a_n$ be real numbers such that $\left|a_i\right|=\left|a_{i-1}+1\right|$ for all $i$, then the A.M. of the numbers $a_1, a_2, a_3, \ldots, a_n$ has the value $A$ where

Given sum of the first $n$ terms of an A.P. is $2 n+3 n^{2}$. Another

The sum of 100 terms of the series $0.9 + 0.09 + 0.009 \ldots$ will be

The sum to $n$ terms of the series $\frac{4}{3} + \frac{{10}}{9} + \frac{{28}}{{27}} + ...$ is

If the sum of the series $3+\frac{1}{4}(3+d)+\frac{1}{4^2}(3+2 d)+\ldots$ to $\infty$ is 8 , then the value of $d$, is

If $4{a^2} + 9{b^2} + 16{c^2} = 2\left( {3ab + 6bc + 4ca} \right)$, where $a,b,c$ are non-zero numbers, then $a,b,c$ are in

The coefficient of ${x^4}$ in the expansion of $\frac{{1 - 2x - {x^2}}}{{{e^{ - x}}}}$ is

A man saves $₹ 200$ in each of the first three months of his service. In each of the subsequent months his saving increases by $₹ 40$ more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

If $m$ is a root of the given equation $\left( {1 - ab} \right){x^2} - \left( {{a^2} + {b^2}} \right)x - \left( {1 + ab} \right) = 0$ and $m$ harmonic means are inserted between $a$ and $b$, then the difference between the last and the first of the means equals

If the $9^{\text {th }}$ term of an AP is 35 and $19^{\text {th }}$ term is 75 , then its $20^{\text {th }}$ term will be

The minimum value of $3{\tan ^2}\theta + 12{\cot ^2}\theta$ is

If the first, second and the last terms of an A.P. are $a, b$, c respectively, then the sum of the A.P. is

The maximum value of the sum of the AP $50,48,46,44, \ldots$ is

The value of ${\left( {0.16} \right)^{{{\log }_{2.5}}\left( {\frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + \ldots \infty } \right)}}$ , is

If the $\left( {p + q} \right)$th term of a geometric series is $m$ and the $\left( {p - q} \right)$th term is n, then the $p$th term is