Sequence of series practice · Mixed difficulty

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

If $p, q, r$ are in $A P$ and $x, y, z$ are in GP, then $x^{q-r} \cdot y^{r-p} \cdot z^{p-q}$ 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

In a GP, if $(m+n)$ th term is $p$ and $(m-n)$ th term is $q$, then its $m$ th term 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

The sum of the series $\left( {1 + 2} \right) + \left( {1 + 2 + {2^2}} \right) + \left( {1 + 2 + {2^2} + {2^3}} \right) + ...\mathrm{upto}\;n$ terms is

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:

The coefficients of ${x^n}$ in the expansion of ${\log _a}\left( {1 + x} \right)$ is

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

If $x,y,z$ are positive integers, then $\left( {x + y} \right)\left( {y + z} \right)\left( {z + x} \right)$ is

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

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

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

The sixth term of an A.P. is equal to 2 the value of the common difference of the A.P. which makes the product ${a_1}{a_4}{a_5}$ least is given by

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

The harmonic mean of two numbers is 4 and the arithmetic and geometric means satisfy the relation $2\;A + {G^2} = 27,$ the numbers are

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

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

$n$th term of the series $1 + \frac{4}{5} + \frac{7}{{{5^2}}} + \frac{{10}}{{{5^3}}} + ...$ will be

The sum of the series : $^{2}+2(d)^{2}+3(6)^{2}+\ldots$ upto 10 terms is :

If the sum to $n$ terms of an A.P. is $3 n^2+5 n$, while $T_m=164$, then value of $m$ is

If ${a^2} + 4{b^2} = 12ab,$ then $\log \left( {a + 2b} \right) =$

Let $a, b$ and c form a GP of common ratior, with $0 < r < 1$. If $a, 2 b$ and $3 c$ form an $A P$, then $r$ equals

The sum of the first ten terms of an $\mathrm{AP}$ is four times the sum of the first five terms, the ratio of the first term to the common difference is