Sequence of series practice · Easy

If $S_n$ denotes the sum of first $n$ term of an A.P., then $\frac{S_{3 n}-S_{n-1}}{S_{2 n}-S_{2 n-1}}$ is equal to

If the sum of all terms of an A.P. obtained by placing $n$ arithmetic means between 5 and 21 is 117 , then $n=$

In an A.P., the first term is 1 and sum of the first $p$ terms is 0, then sum of the first $(p+q)$ terms is

Suppose $a_1, a_2, \ldots a_n, \ldots$ be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is $5: 17$ and $110\, < \,{a_{15}}\, < \,120$, then the sum of the first ten terms of the progression is equal to

Let $a_1, a_2, a_3, \ldots, a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,3, \ldots, 10$ and $S$ be the set of pairs $(r, k), r, k \in N$ for which
$D=\left|\begin{array}{lll}\log _e a_1^r a_2^k & \log _e a_2^r a_3^k & \log _e a_3^r a_4^k \\ \log _e a_4^r a_5^k & \log _e a_5^r a_6^k & \log _e a_6^r a_7^k \\ \log _e a_7^r a_8^k & \log _e a_8^r a_9^k & \log _e a_9^r a_{10}^k\end{array}\right|=0$
Then the number of elements in $S$, is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the common ratio of this progression equals

The product of $x^{1 / 2} \cdot x^{1 / 4} \cdot x^{1 / 8}$.. Equals

If three positive real numbers $a,\;b,\;c$ are in AP and $abc = 4$, then the minimum possible value of $b$ is

Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is 4 and the second term is $3 / 4$, then

If $5 \log _{a b c}\left(a^3+b^3+c^3\right)=3 \lambda\left(\frac{1+\log _3(a b c)}{\log _3(a b c)}\right)$ and $(a b c)^{a+b+c}=1$ and $\lambda=\frac{m}{n}$, where $m$ and $n$ are relative primes, the value of $|m+n|+|m-n|$ is

If S is the sum to infinity of a G.P., whose first term is $a,$ then the sum of the first $n$ terms is

For a sequence $< {a_n} > ,{a_1} = 2$ and $\frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{1}{3}.$ Then, $\sum \limits_{r = 1}^{20} {a_r}$ is

Every term of a G.P. is positive and also every term is the sum of two preceding terms. Then, the common ratio of the G.P. is

If $a_1, a_2, a_3, \ldots, a_n$ are in H.P. then $\frac{a_1}{a_2+a_3+\ldots+a_n}, \frac{a_2}{a_1+a_3+\ldots+a_n}, \frac{a_3}{a_1+a_2+a_4+\ldots+a_n}$, $\ldots, \frac{a_n}{a_1+a_2+\ldots+a_{n-1}}$ are in

The GM of roots of the equation ${x^2} - 18x + 9 = 0$ is

Let $x \in \left( {1,\infty } \right)$ and $n$ be a positive integer greater than 1. If ${f_n}\left( x \right) = \frac{n}{{\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_3}x}} + \ldots + \frac{1}{{{{\log }_n}x}}}}$, then ${\left( {n!} \right)^{{f_n}\left( x \right)}}$ equals to

If $a_1, a_2, a_3, \ldots, a_n$ are in A.P., where $a_i>0$ for all $i$, then the value of
$\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}$, is

The sum of $n$ terms of the following series $1 + \left( {1 + x} \right) + \left( {1 + x + {x^2}} \right) + \ldots$ is

The sum of the series ${\log _4}2 - {\log _8}2 + {\log _{16}}2 - \ldots ,$ is

If $\frac{{a + b}}{{1 - ab}},b,\frac{{b + c}}{{1 - bc}}$ are in A.P., then $a,\frac{1}{b},c$ are in

If $\sqrt{3}, A$ and $\sqrt{2}$ are in $A P$, then $\frac{\sqrt{3}+\sqrt{2}}{2}$ is greater than or equal to

In an increasing A.P., the sum of three consecutive terms is 51 and the product of first and third terms is 273. Its third term is

If $f(n)=\sum\limits_{{r=1}}^n r^4$, then the value of $\sum\limits_{{r=1}}^n r(n-r)^3$ is equal to

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

The third term of an AP is 18 and the seventh term is 30 , then the sum of the first 17 terms is

If $a,b,c$ are in H.P., then $\frac{a}{{b + c}},\frac{b}{{c + a}},\frac{c}{{a + b}}$ will be in

If $\alpha ,\,\beta$ are natural numbers, such that ${100^\alpha } - 199\beta = (100)(100) + (99)(101) + (98)(102) + .... + (1)(199)$, then the slope of the line passing through $(\alpha ,\,\beta )$ and origin is

If the sum to $2n$ terms of the AP 2, 5, 8, 11,… is equal to the sum to n terms of the AP 57, 59, 61, 63,…, then $n$ is equal to

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

If $x = 1 + a + {a^2} + ...\infty$ and $y = 1 + b + {b^2} + ...\infty$ where $a$ and $b$ are proper fractions, then $1 + ab + {a^2}{b^2} + ...\infty$ equals