Sequence of series practice · Easy

$\sum\limits_{{j=1}}^{n} \sum\limits_{{i=1}}^{n} i$ is equal to

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternativety positive and negative, then the first term is

The ratio of the sum of first three terms of a G.P. to the sum of first 6 terms of the G.P. is $125: 152$, then common ratio of the G.P. is

The sets ${S_1},{S_2},{S_3} \ldots$ are given by ${S_1} = \left\{ {\frac{2}{1}} \right\},{S_2} = \left\{ {\frac{3}{2},\frac{5}{2}} \right\},{S_3} = \left\{ {\frac{4}{3},\frac{7}{3},\frac{{10}}{3}} \right\},{S_4} = \left\{ {\frac{5}{4},\frac{9}{4},\frac{{13}}{4},\frac{{17}}{4}} \right\}, \ldots$ Then, the sum of the numbers in the set ${S_{25}}$ is

Let the sequence $a_1, a_2, a_3, \ldots, a_n, \ldots$ form an A.P. Then, $a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2 n-1}^2-a_{2 n}^2$ is equal to

Let ${S_n}$ be the sum of the first $n$ terms of an arithmetic progression, If ${S_{3n}} = 3{S_{2n}}$, then the value of $\frac{{{S_{4n}}}}{{{S_{2n}}}}$ is

If 17 arithmetic means are inserted between $7 / 2$ and $-83 / 2$, then the 17 th $\mathrm{AM}$ is

Let $a_{1}, a_{2}, a_{3} \ldots \ldots \ldots .$. be terms on A.P. If $\frac{a_{1}+a_{2}+\ldots \ldots \ldots . . a_{p}}{a_{1}+a_{2}+\ldots \ldots \ldots . .+a_{q}}=\frac{p^{2}}{q^{2}}, p \neq q$, then $\frac{a_{6}}{a_{21}}$ equals

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

If the A.M. and G.M. of two numbers $a$ and $b$ are 2 and 4 respectively, their H.M. is

Let $\left\{a_n\right\}_{n=0}^{\infty}$ be a sequence such that $a_0=a_1=0$ and $a_{n+2}=3 a_{n+1}-2 a_n+1, \forall n \geq 0$
Then $a_{25}\, a_{23}-2 a_{25}\, a_{22}-2 a_{23}\, a_{24}+4 a_{22}\, a_{24}$ is equal to

If ${H_1},{H_2}$ are two harmonic means between two positive numbers $a$ and $b\;\left( {a \ne b} \right)$, $A$ and $G$ are the arithmetic and geometric means between $a$ and $b$, then $\frac{{{H_2} + {H_1}}}{{{H_2}{H_1}}}$ is

Let $a, b \in R^{+}$, such that $\log _{27} a+\log _9 b=\frac{7}{2}$ and $\log _{27} b+\log _9 a=\frac{2}{3}$, then $a b$ is equal to:

If $\sum\limits_{{i=1}}^{21} a_i=693$, where $a_1, a_2, \ldots, a_{21}$ are in $A P$, then the value of $\sum\limits_{{i=0}}^{10} a_{2 r+1}$ is

Statement -1 The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16)$ $+\ldots+(361+380+400)$ is 8000 .

Statement $2 \sum_{k=1}^n\left(k^3-(k-1)^3\right)=n^3$ for any natural number $n$.

Sum of the series $S=1^2-2^2+3^2-4^2+$ $\ldots-2008^2+2009^2$ is

The sum of the intercepts cut off by the axes on the lines $x+y=a, x+y=a r, x+y=a r^2, \ldots$ where $a \neq 0$ and $r=1 / 2$ is

The sum of the series $2\left\{ {{7^{ - 1}} + {3^{ - 1}} \cdot {7^{ - 3}} + {5^{ - 1}} \cdot {7^{ - 5}} + \ldots } \right\}$ is

If the $n$th terms of two series $3+10+17+\ldots$ and $63+65+67+\ldots$ are equal, then $n=$

A GP consists an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio will be equal to

If $x=1+\log _a b c, y=1+\log _b c a, z=1+\log _c a b,$ then $\frac{x y z}{x y+y z+z x}$ is equal to

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

The sum of $i - 2 - 3i + 4 \ldots .$ upto 100 terms, where $i = \sqrt { - 1}$ is

Suppose $a_1, a_2, \ldots$ are in A.P. If $a_8: a_5=$ $3: 2$, then $a_{17}: a_{23}$ is:

A species has an initial population $4^{10}$. At the end of first day, the population increases by $50 \%$. At the end of second day, it decreases by the same percentage. If the process continues in the same pattern, the number of days for the population to reach $3^{10}$ is

If AM of two terms is 9 and HM is 36 , then $G M$ will be

If the ratio of sum of $n$ terms of two A.P's is $(3 n+8):(7 n+15)$, then the ratio of $12^{\text {ti }}$ terms is-

The sum of the series $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + 3 \cdot 4 \cdot 5 + ...$ to $n$ terms is

If $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots$ upto $\infty=\frac{\pi^2}{6}$, then value of $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots$ upto $\infty$ is

The AM, HM and GM between two numbers are $\frac{{144}}{{15}},\;15$ and 12, but not necessarily in this order. Then, HM, GM and AM respectively are