Sequence of series practice · Mixed difficulty

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 the sequence $< a_n >$ is defined by $a_n=3 n+2$, then difference of any two consecutive terms is

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

The sum of n terms of the series $1 + \left( {1 + x} \right) + \left( {1 + x + {x^2}} \right) + \;...$ will be

If $a^x=b^y=c^z$ and $a, b, c$ are in GP, then $x, y$ and $=$ are in

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

The sum of first four terms of a GP is $12(1-\sqrt{5})$ If the common ratio is $-\sqrt{5}$, then the first term of a GP is

If $x,y,z$ be three positive prime numbers. The progression in which $\sqrt x ,\sqrt y ,\sqrt z$ can be three terms (not necessarily consecutive) is

The first and last terms of an AP are 1 and 11 . If the sum of its terms is 36 , then the number of terms will be

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

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

The sum of the first 20 terms of the series $5+11+$ $19+29+41+\ldots$ is

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 $a$ and $b$ are two positive numbers, then

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

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

If the sum of first $n$ terms of a series is $5 n^2+2 n$, then its second term is

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

If ${\log _2}a + {\log _4}b + {\log _4}c = 2$

${\log _9}a + {\log _3}b + {\log _9}c = 2$

${\log _{16}}a + {\log _{16}}b + {\log _4}c = 2$ then

The arithmetic mean, geometric mean and harmonic mean between two positive real quantities are themselves in

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

The series of natural numbers is divided into groups $(1), (2,3,4),(5,6,7,8,9)$, and so on. The sum of first and last term of the $n$th group is

$\left(\frac{a-b}{a}\right)+\frac{1}{2}\left(\frac{a-b}{a}\right)^2+\frac{1}{3}\left(\frac{a-b}{a}\right)^3+\ldots=$ ?

If $f(x)$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x=1}^n f(x)=120$. Then, the value of $n$ is

Suppose A.M. of two positive numbers be 7 and G.M. between them be 5. The A.M. between their squares is

If H.M. : G.M. $=4: 5$ for the two positive numbers, then the ratio of the numbers is

The solution of the equation ${\log _\mathrm{\pi }}\left( {{{\log }_2}\left( {{{\log }_7}x} \right)} \right) = 0,$ is