Sequence of series practice · Medium

The sum of numbers of three digits which are divisible by 7 , is

Let $a$ and $b$ be two positive real numbers. Suppose $A_1$, $A_2$ are two arithmetic means; $G_1, G_2$ are two geometric means and ${H}_1, {H}_2$ are two harmonic means between $a$ and $b$, then $\frac{A_1+A_2}{H_1+H_2}$ is equal to

A person purchases $1 \mathrm{~kg}$ tomatoes from each of the 4 places at the rate of $1 \mathrm{~kg}, 2 \mathrm{~kg}, 3 \mathrm{~kg}$ and $4 \mathrm{~kg}$ per rupee respectively. On the average, he has purchased $x \mathrm{~kg}$ tomatoes per rupee, then the value of $x$ is

The value of $\lim _{r \rightarrow-\infty} \sum_{r=1}^n \tan ^{-1}\left(\frac{1}{2 r^2}\right)$ is

If $a,{a_1},{a_2}, \ldots ,{a_{2n}},b$ are in arithmetic progression and $a,{g_1},{g_2}, \ldots ,{g_{2n}},b$ are in geometric progression and $h$ is the harmonic mean of $a$ and $b$, then $\frac{{{a_1} + {a_{2n}}}}{{{g_1}{g_{2n}}}} + \frac{{{a_2} + {a_{2n - 1}}}}{{{g_2}{g_{2n - 1}}}} + ... + \frac{{{a_n} + {a_{n + 1}}}}{{{g_n}{g_{n + 1}}}}$ is equal to

A man wants to cut three lengths from a single piece of board of length $91 \mathrm{~cm}$. The second length is to be $3 \mathrm{~cm}$ longer than the shortest and the third length is to be twice as long as the shortest. The possible length of the shortest board, if the third piece is to be at least $5 \mathrm{~cm}$ longer than the second, is

In a certain AP, 5 times the 5 th term is equal to 8 times the 8 th term, then 13 th term is

The value of $\sum\limits_{{k=1}}^n(n-k) \cos \frac{2 k \pi}{n} \, (n \geq 3)$ is

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x))=x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \ldots, a_n$ be in arithmetic progression mean is zero, then the value of $f\left(g\left(\frac{1}{n} \sum\limits_{i=1}^n f\left(a_i\right)\right)\right)$ is equal to

An infinite GP has the first term $'x'$ and sum 5, then $x$ belongs to

the sum of $n$ terms of the infinite series

$1 \cdot {3^2} + 2 \cdot {5^2} + 3 \cdot {7^2} + ...\infty$ is

If $S_n=n P+\frac{n}{2}(n-1) Q$, where $S_n$ denotes the sum of first $n$ terms of an AP, then the common difference is

Given two numbers $a$ and $b$. Let $A$ denote the single A.M. and $S$ denote the sum of $n$ A.M.’s between $a$ and $b$, then $S/A$ depends on

If $x>0$ and $\log _2 x+\log _2(\sqrt{x})+\log _2(\sqrt[4]{x})$ $+\log _2(\sqrt[8]{x})+\log _2(\sqrt[16]{x})+\ldots=4$, then $x$ equals

Let ${a_1},{a_2}, \ldots ,{a_{10}}$ be in AP and ${h_1},{h_2}, \ldots ,{h_{10}}$ be in HP. If ${a_1} = {h_1} = 2\mathrm{\;and\;}{a_{10}} = {h_{10}} = 3$ then ${a_4}{h_7}$ is

If $\left(a^{m}\right)^{n}=a^{m^{n}}$, then the value of ' $\mathrm{m}^{\prime}$ in terms of ' $\mathrm{n}$ ' is

Let $b_1, b_2, \ldots \ldots \ldots .$. , be a geometric sequence such that $b_1+b_2=1$ and $\sum\limits_{{k=1}}^{\infty} b_k=2$. Given that $b_2<0$, then the value of $b_1$ is:

If the sum of $n$ terms of an AP is $3 n^2-n$ and its common difference is 6 , then its first term is

If $a, b$ and $c$ are in HP, then correct statement is

If the arithmetic mean and geometric mean of $a$ and $b$ are $A$ and $G$ respectively, then the value of $A-G$ will be

If $a, b$ and $c$ are in $\mathrm{AP}$, then $(a+2 b-c)(2 b+c-a)$ $(c+a-b)$ equals

Let $p_n$ denote the probability of getting $n$ heads when a fair coin is tossed $m$ times. If $P_4, P_5, P_6$ are in Arithmetic Progression, then the sum of all possible values of $m$ is

The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^{\circ}$. Number of sides of the polygon if its largest interior angle is $172^{\circ}$ are

The natural numbers are written as follows

$\begin{array}{cccccccccccccccccccc} &&&1 \\ &&2&&3& \\ &4&&5&&6 \\ 7&&8&&9&&{10} \\ \begin{gathered} . \\ . \\ . \\ \end{gathered} &&\begin{gathered} . \\ . \\ . \\ \end{gathered} &{}&\begin{gathered} . \\ . \\ . \\ \end{gathered} &{}&\begin{gathered} . \\ . \\ . \\ \end{gathered} \end{array}$

The sum of the numbers in the $n^{\text {th}}$ row is

If $a, a_1, a_2, a_3, \ldots, a_{2 n}, b$ are in AP and $a, g_1, g_2, \ldots, g_{2 n}, b$ are in GP and $h$ is the HM of $a$ and $b$, then $\frac{a_1+a_{2 n}}{g_1 g_{2 n}}+\frac{a_2+a_{2 n-1}}{g_2 g_{2 n-1}}+\ldots+\frac{a_n+a_{n+1}}{g_n g_{n+1}}$ is equal to

$1.2 .3+2.3 .4+3.4 .5+\ldots \ldots+\mathrm{n}$ terms -

If the sum of $m$ terms of an A.P. is the same as the sum of its $n$ terms, then the sum of its $(m+n)$ terms is .

If $p$th, $q$th and $r$th terms of a G.P. are $a, b, c;$ then $\left(\frac{c}{b}\right)^p\left(\frac{b}{a}\right)^r\left(\frac{a}{c}\right)^q=$

In an increasing geometric series, the sum of the second and the sixth term is $\frac{{25}}{2}$ and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to

If $\log _3 2, \log _3\left(2^x-5\right)$ and $\log _3\left(2^x-7 / 2\right)$ are in AP, then $x$ is equal to