Sequence of series practice · Mixed difficulty

Let $a_1, a_2, \ldots a_n, \ldots$ be a G.P. of positive terms such that $\sum\limits_{n=1}^{100} a_{2 n+1}=200, \sum\limits_{n=1}^{100} a_{2 n}=100$. Then, $\sum\limits_{n=1}^{200} a_n$ is equal to

If $a_1, a_2, a_3, a_4, a_5$ are in $\mathrm{HP}$, then $a_1 a_2+a_2 a_3+a_3 a_4+a_4 a_5$ is equal to

Three positive numbers form an increasing GP. If the middle terms in this GP is doubled, the new numbers are in AP. Then, the common ratio of the GP is

If $G$ is the GM of the product of $r$ set of observation with geometric means ${G_1},\;{G_2}, \ldots {G_r}$ respectively, then $G$ is equal to

If $a,b,c$ are in H.P., then $\frac{1}{{b - a}} + \frac{1}{{b - c}} =$

If the sum of $n$ terms of the series

$1 + \frac{4}{5} + \frac{7}{{{5^2}}} + \frac{{10}}{{{5^3}}} + \ldots$ is $l + \frac{{15}}{{16}}\left( {1 - \frac{1}{{{5^{n - 1}}}}} \right) - \frac{{\left( {3n - 2} \right)}}{{4\left( {{5^{n - 1}}} \right)}}$, then $l$ is

The sum of first 26 terms of an A.P. $a_1, a_2, a_3, \ldots$ if $a_2+a_6+a_9+a_{18}+a_{21}+a_{25}=165$, is

If sum of four numbers in A.P. is 28 and product of two middle terms is 45, then product of the first and last terms is

If $\log 4, \log \left(2^x+2\right), \log \left(2^{x+2}+1\right)$ are 3 consecutive terms of an arithmetic progression then $x$ lies between 2 consecutive natural numbers $p$ and $q(p < q)$ find the value of $(p+q)$.

The sum of the series $8 !+\frac{9 !}{2 !}+\frac{10 !}{3 !}+\frac{11 !}{4 !}+\ldots$. up to 20 terms

If ${\log _y}x = {\log _z}y = {\log _x}z,$ then

If arithmetic mean of two positive numbers is $A$, their geometric mean is $G$ and harmonic mean is $H$, then $H$ is equal to

The maximum sum of an AP $40,38,36,34, \ldots$ is

If the first and the ${n}^{\text {th }}$ term of $a$ G.P. are $a$ and $b$ respectively and $P$ is the product of the first $n$ terms, then $P^2=$

If the first, second and last terms of an arithmetic series are $a,b$ and $c$respectively, then the number of terms is

Let the sum of the first $n$ terms of a non-constant AP ${a_1},\,{a_2},\,{a_3}....\,\,$ be $50n + \frac{{n(n - 7)}}{2}A,\,$ where $A$ is a constant. If $d$ is the common difference of this AP, then the ordered pair $(d,\,{a_{50}})$ is equal to

If $\frac{{\log x}}{{{a^2} + ab + {b^2}}} = \frac{{\mathrm{log}y}}{{{b^2} + bc + {c^2}}} = \frac{{\log z}}{{{c^2} + ca + {a^2}}},$ then ${x^{a - b}}.{y^{b - c}}.{z^{c - a}} =$

The coefficient of ${x^3}$ in the expansion of ${3^x}$ is

In a sequence of 21 terms, the first 11 terms are in AP with common difference 2 and the last 11 terms are in GP with common ratio 2. If the middle term of AP be equal to the middle term of the GP, then the middle term of the entire sequence is

If every even term of a series is a times the term before it and every odd term is $c$ times the term before it, the first term being unity, then the sum to $2 n$ terms is

Let $a_n$ be the nth term an A.P. If $\sum\limits_{r=1}^{100} a_{2 r}=\alpha$ and $\sum\limits_{r=1}^{100} a_{2 r-1}=\beta$, then the common difference of the A.P., is

Suppose $a,\;b,\;c$ are in AP and ${a^2},\;{b^2},\;{c^2}$ are in GP. If $a < b < c$ and $a + b + c = \frac{3}{2}$, then the value of $a$ is

$\left\{ {{a_n}} \right\}\;and\;\left\{ {{b_n}} \right\}$ be two sequences given by ${a_n} = {\left( x \right)^{\frac{1}{{{2^n}}}}} + {\left( y \right)^{\frac{1}{{{2^n}}}}}$ and ${b_n} = {\left( x \right)^{\frac{1}{{{2^n}}}}} - {\left( y \right)^{\frac{1}{{{2^n}}}}}$ for all $n \in N,$ then ${a_1}{a_2}{a_3} \ldots {a_n}$ is equal to

The $n$th term of the sequence $2 \frac{1}{2}, 1 \frac{7}{13}, 1 \frac{1}{9}, \frac{20}{23} \ldots$ is

If the sum of the first 2 n terms of the A.P. 2,5,8, is equal to the sum of the first $n$ terms of the A.P. 57, 59, 61,...., then $n$ equals

Let $a_1, a_2, a_3, \ldots, a_n, \ldots \ldots .$. be a G.P such that $\frac{a_4}{a_6}=\frac{1}{4}$ and $a_2+a_5=216$. Then, $a_1=$

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

If $x=1+y+y^2+\ldots$, then $y$ is

The geometric mean of the numbers $3,3^2, 3^3, \ldots, 3^n$ is

The number of terms of the A.P. $1, 4, 7, \ldots$ that must be taken to obtain a sum of 715 is