Sequence of series practice · Easy

If $\sum\limits_{{r=1}}^n T_r=\frac{n(n+1)(n+2)(n+3)}{8}$, then $\lim \limits_{{n \rightarrow \infty}} \sum\limits_{{r=1}}^n \frac{1}{T_r}=$

If ${\log _e}2.{\log _x}27 = {\log _{10}}8.{\log _e}10,$then $x =$

If ${a_1},\;{a_2}, \ldots ,\;{a_{n + 1}}$ are in AP, then $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + ... + \frac{1}{{{a_n}{a_{n + 1}}}}$ is

The $9^{\text {th }}$ term of the series $27+9+5 \frac{2}{5}+3 \frac{6}{7}+\cdots$ will be

If sum of the first 2$n$ terms of an AP series 2, 5, 8,… is equal to the sum of the first $n$ terms of the AP series 57, 59, 61,…, then $n$ equals

If $\tan \left( {\frac{\pi }{9}} \right),\,x,\,\tan \left( {\frac{{7\pi }}{{18}}} \right)$ are in arithmetic progression and $\tan \left( {\frac{\pi }{9}} \right),\,y,\,\tan \left( {\frac{{5\pi }}{{18}}} \right)$ are also in arithmetic progression, then $\left| {x - 2y} \right|$ is equal to

The sum of the series

upto 15 terms is

A person is to count 4500 currency notes. Let $a_n$ denote the number of notes he counts in the $n$th minute. If $a_1=a_2=\ldots=a_{10}=150$ and $a_{10}, a_{11}, \ldots$ are in AP with common difference - 2, then the time taken by him to count all notes is

If $a,b,c$ are in A.P., $b - a,c - b$ and a are in G.P., then $a:b:c$ is

If $9{a^2} + 4{b^2} = 18ab,$ then $\log \left( {3a + 2b} \right) =$

If $x=1998$ !, then value of the expression $\frac{1}{\log _{2} x}+\frac{1}{\log _{3} x}$ $+\ldots \ldots+\frac{1}{\log _{1998} \mathrm{x}}$ equals-

If $x,y,z$ are in H.P., then $\log \left( {x + z} \right) + \log \left( {x - 2y + z} \right)$ is equal to

0.14189189189… can be expressed as a rational number

If $\frac{1}{a} + \frac{1}{c} + \frac{1}{{a - b}} + \frac{1}{{c - b}} = 0$ and $b \ne a + c,$ then $a,b,c$ are in

Let $a_1, a_2, a_3, \ldots$ be a harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for which $a_n<0$, is

Let $f\left( x \right)$ be a polynomial function of second degree. If $f\left( 1 \right) = f\left( { - 1} \right)$ and $a,b,c$ are in AP, then $f'\left( a \right),\;f'\left( b \right)$ and $f'\left( c \right)$ are in

If $3^{\text {rd }}, 6^{\text {th }} \&$ last term of HP is $\frac{1}{3}, \frac{1}{5}, \frac{3}{203}$. Then the number of terms is

If the function $f: R \rightarrow R$ satisfies $f(x+y)=f(x)+f(y), \forall \, x, y \in \mathrm{R}$ and $f(1)=7$, then $\sum\limits_{{r=1}}^n f(r)$

The sum of the series $\frac{1}{{x + 1}} + \frac{2}{{{x^2} + 1}} + \frac{{{2^2}}}{{{x^4} + 1}} + .... + \frac{{{2^{100}}}}{{{x^{{2^{100}}}} + 1}}\,$, when $x = 2$ is

If the sum of an infinite GP and the sum of square of its term is 3, then the common ratio of the first series is

If $1^3+2^3+3^3+\ldots+n^3=2025$, then $n=$

Consider the following statements.

I. A sequence is called an arithmetic progression if the difference of a term and the previous term is always same.

II. Arithmetic Mean (A.M.) $A$ of any two numbers $a$ and $b$ is given by $\frac{1}{2}(a+b)$ such that $a, A, b$ are in A.P. The arithmetic mean for any $n$ positive numbers $a_1, a_2, a_3, \ldots ., a_n$ is given by

$\text {A.M. }=\frac{a_1+a_2+\ldots .+a_n}{n}$

Choose the correct option.

If ${\log _8}x = 2.5$ and ${\log _8}x = 2.5$ then $x =$

Let $S = \frac{8}{5} + \frac{{16}}{{15}} + ... + \frac{{128}}{{{2^{18}} + 1}},$, then

The first term of a GP is 7, the last term is 448 and sum of all term is 889, then the common ratio is

The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\ldots$ is

If $a,b,c$ be in arithmetic progression, then the value of $\left( {a + 2b - c} \right)\left( {2b + c - a} \right)\left( {a + 2b + c} \right)$ is

Let $a_1, a_2, \ldots, a_9$ be in G.P. with $a_1 < 0$ such that $a_1+a_2=4, a_3+a_4=16$. If $\sum\limits_{i=1}^9 a_i=4 \lambda$, then $\lambda$ is equal to

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

The value of ${\log _3}e - {\log _9}e + {\log _{27}}e - {\log _{81}}e + \ldots \mathrm{ad}.\mathrm{inf}.,$ is