Sequence of series practice · Easy

The sum of the series $\frac{1}{{\sqrt 1 + \sqrt 2 }} + \frac{1}{{\sqrt 2 + \sqrt 3 }} + \frac{1}{{\sqrt 3 + \sqrt 4 }} + \ldots + \frac{1}{{\sqrt {{n^2} - 1} + \sqrt {{n^2}} }}$ equals

GM and HM of two numbers are 10 and 8 respectively. The numbers are

If $\frac{1}{{{{\log }_3}\mathrm{\pi }}} + \frac{1}{{{{\log }_4}\mathrm{\pi }}} > x,$ then the greatest integral value of $x$ is

If 7th term of an A.P. is 9 and 9th term of the A.P. is 7, then 20th term of the A.P. is

If sum of the infinite G.P. $p + 1 + \frac{1}{p} + \frac{1}{{{p^2}}} + .....,\,\left( {p > 2} \right)$ is $49/6,$ then sum of the 3rd term and the 4th term of the G.P. is

If the sum of the first 20 terms of the series $\log _7 1 / 2+\log _7 1 / 3 x+\log _7 1 / 4 x+\ldots$ is 460 , then $x$ is equal to

If $1 + \lambda + {\lambda ^2} + ... + {\lambda ^n} = \left( {1 + \lambda } \right)\left( {1 + {\lambda ^2}} \right)\left( {1 + {\lambda ^4}} \right)\left( {1 + {\lambda ^8}} \right)\left( {1 + {\lambda ^{16}}} \right),$ then the value of $n$ is (where $n \in N)$

The digits of a three digit number $N$ are in A.P. If sum of the digits is 15 and the number obtaind by reversing the digits of the number is 594 less than the original number, then $\frac{1000}{N-252}$ is equal to

If the arithmetic progression whose common difference is non-zero, the sum of first $3 n$ terms is equal to the sum of the next $n$ terms. The ratio of the sum of the first $2 n$ terms to the next $2 n$ terms is

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b+2$, then the value of $\frac{a^2+a-14}{a+1}$, is

If $x=\log _5(1000)$ and $y=\log _7(2058),$ then

If $p,q,r$ are in GP and ${\tan ^{ - 1}}p,{\tan ^{ - 1}}q,{\tan ^{ - 1}}r$ are in AP, then $p,q,r$ satisfies the relation

If the sum of first $p$ terms, first $q$ terms and first $r$ terms of an A.P. be $x,y$ and $z$ respectively. Then, $\frac{x}{p}\left( {q - r} \right) + \frac{y}{q}\left( {r - p} \right) + \frac{z}{r}\left( {p - q} \right)$ is

If $\log \left( {x + z} \right) + \log \left( {x - 2y + z} \right) = 2\log \left( {x - z} \right),$ then $x,y,z$ are in

The number of real values of the parameter $\lambda$ for which $\left(\log _{16} x\right)^2-\log _{16} x+\log _{16} \lambda=0$ with real coefficients will have exactly one solution is

If the sum of four numbers of $A P$ is 20 , then which of the following is first term of the given $A P$ ?

The sum of three numbers in A.P. is 21. If
the second number is reduced by 1 while third is increased
by 1, three resulting numbers form an increasing G.P., the
common ratio of the G.P. is

If one geometric $G$ and two arithmetic means $A_1$ and $A_2$ are inserted between two distinct positive
numbers, then $\left( {\frac{{2{A_1} - {A_2}}}{G}} \right)\left( {\frac{{2{A_2} - {A_1}}}{G}} \right)$ is equal to

The sum to $n$ terms of the series $\frac{3}{2}+\frac{5}{4}+\frac{9}{8}+\frac{17}{16}+\ldots$ is equal to

In a G.P. of positive terms, any term is equal to the sum of the next two terms. Then, the common ratio of the G.P. is

For any three positive real numbers $a, b$ and $c$, $9\left(25 a^2+b^2\right)+25\left(c^2-3 a c\right)=15 b(3 a+c)$. Then,

If the sum of $n$ terms of a AP is $nA + {n^2}B,$ where $A$ and $B$ are constants, then its common difference will be

If ${4^{{{\log }_9}3}} + {9^{{{\log }_2}4}} = {10^{{{\log }_x}83}},$ then $x =$

$7^{\text {th }}$ term of an A.P. is 40 , then the sum of first 13 terms is

If $a=0.2, b=\sqrt{5}$ and $x=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots \infty$, then $a^{\log _b x}=$

$1 + \frac{{{{\left( {{{\log }_e}n} \right)}^2}}}{{2!}} + \frac{{{{\left( {{{\log }_e}n} \right)}^4}}}{{4!}} + ...$ is equal to

If ${x^{18}} = {y^{21}} = {z^{28}},$ then $3,3{\log _y}x,\;3{\log _z}y,\;7{\log _x}z$ are in

The first two terms of a geometric progression add upto 12 the sum of the third and the fourth terms is 48 , if the terms of the geometric progression are alternately positive and negative, then the first term is

The sequence $S=i+2 i^2+3 i^3+\ldots .$. upto 100 terms simplifies to where $\mathrm{i}=\sqrt{-1}$

If $n$ is even, then the sum of $n$ terms of the series $1^2-2^2+3^2-4^2+5^2-6^2+\ldots \ldots$, is