Sequence of series practice · Medium

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

The sum of the series ${1.3^2} + {2.5^2} + {3.7^2} + \;...$ upto 20 terms is

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

The sum of first $n$ terms of the series ${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + \ldots$ is $\frac{{n{{\left( {n + 1} \right)}^2}}}{2}$ when $n$ is even. When $n$ is odd the sum is

If $x_1, x_2, \ldots, x_n$ are $n$ non-zero real numbers such that $\begin{aligned}\left(x_1^2+x_2^2+\ldots+x_{N-1}^2\right) & \left(x_2^2+x_3^2+\ldots+x_N^2\right) \\ & \leq\left(x_1 x_2+x_2 x_3+\ldots+x_{n-1} x_n\right)^2,\end{aligned}$ then $x_1, x_2, \ldots, x_n$ are in

Between two numbers whose sum is $2 \frac{1}{6}$, an even number of arithmetic means is inserted; the sum of these means exceeds their number by unity. Then, the number of means is

If $5,25,125, \ldots$ are in $G P$, then $1,5,25, \ldots$ are in

The sum to $n$ terms of the series $11+103+1005+\ldots$. Is

In the four numbers first three are in GP and last three are in AP whose common difference is 6. If the first and last numbers are same, then first number will be

The sum of $n$ terms of the series $5+7+13+31+85+\ldots$ is

If the numbers $a, b, c, d, e$ form an A.P., then the value of $a-4 b+6 c-4 d+e$, is

In a sequence of $(4 n+1)$ terms the first $(2 n+1)$ terms are in A.P. whose common difference is 2, and the last $(2 n+1)$ terms are in G.P. whose common ratio is 0.5. If the middle terms of the A.P. and G.P. are equal, then the middle term of the sequence is

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

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

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

$\sum_{r=1}^n \frac{r^2-r-1}{(r+1) !}$ is equal to

If $n$ is an odd integer greater than or equal to 1, then the value of $n^3-(n-1)^3+(n-2)^3-\ldots .+(-1)^{n-1} 1^3$, 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

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

Let $k$ be natural number. Defined $S_k$ as the sum of the infinite geometric series with first term $\left(k^2-1\right)$ and common ratio $\frac{1}{k}$, that is $S_k=\frac{k^2-1}{k^0}+\frac{k^2-1}{k^1}+\frac{k^2-1}{k^2}+\ldots \ldots$. The value of $\sum\limits_{{k=1}}^{\infty} \frac{S_k}{2^{k-1}}$, is:

If $\log _6 a+\log _6 b+\log _6 c=6,$ where $a, b, c \in N$ and $a, b, c$ are in GP and $b-a$ is a square of an integer, then the value of $a+b-c$ is

If $1,{\log _9}\left( {{3^{1 - x}} + 2} \right),{\log _3}\left( {{{4.3}^x} - 1} \right)$ are in A.P., then $x$ equals

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

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

If $a,b,c,$ are in A.P.; $b,c,d$ are in G.P. $c,d,e,$ are in H.P., then $a,c,e,$ are in

Three distinct numbers $a_1, a_2, a_3$ are in increasing G.P. $a_1^2+a_2^2+a_3^2=364$ and $a_1+a_2+a_3=26$, then the value of $a_{10}$ if $a_n$ is the $n^{\text {th }}$ term of the given G.P. 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

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