Arithmetic progression

If $n$ arithmetic means are inserted between $a$ and 100 such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is :

Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______.

Let a, b be two non-zero real numbers. If p and r are the roots of the equation $x^2-8ax+2a=0$ and q and s are the roots of the equation $x^2+12bx+6b=0$, such that $\dfrac1p,\dfrac1q,\dfrac1r,\dfrac1s$ are in A.P., then $a^{-1}-b^{-1}$ is equal to ______.

The series of positive multiples of 3 is divided into sets : {3}, {6, 9, 12}, {15, 18, 21, 24, 27}, ... Then the sum of the elements in the 11$^{\text{th}}$ set is equal to ______.

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.

Suppose $a_1,a_2,\ldots,a_n,\ldots$ be an arithmetic progression of natural numbers. If the ratio 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 the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be 5. Let the sum of its first five terms be $\dfrac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10ar$, $n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$, is equal to :

Let $\alpha$, $\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha x^5+\beta x^3+\gamma x$, $x\in\mathbf R$ and $g:\mathbf R\to\mathbf R$ be such that $g(f(x))=x$ for all $x\in\mathbf R$. If $a_1,a_2,a_3,\ldots,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\dfrac1n\displaystyle\sum_{i=1}^nf(a_i)\right)\right)$ is equal to :

For $p,q\in\mathbf R$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf R$ and $q>0$. Let $a_1$, $a_2$, $a_3$ and $a_4$ be in an arithmetic progression with mean p and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ______.

Let the coefficients of the middle terms in the expansion of $\left(\dfrac1{\sqrt6}+\beta x\right)^4$, $(1-3\beta x)^2$ and $\left(1-\dfrac\beta2x\right)^6$, $\beta>0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P., then $50-\dfrac{2d}{\beta^2}$ is equal to ______.

For |α| ≥ 1, let $5^{4-2\alpha}$, 63, $5^{2\alpha-1}$ be in A.P. with the common difference d. Let S$_n$ denote the sum of the first n terms of an A.P., whose first term is α$^2$ and the common difference is d. If S$_{30}$ − S$_{15}$ = 30k, then k is equal to

Let A = {n ∈ ℕ : n is a 3-digit number}

B = {9k + 2 : k ∈ ℕ}

and C = {9k + l : k ∈ ℕ} for some l (0 < l < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then l is equal to _______.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_1 = b_1 = c$, then the number of all possible values of $c$, for which the equality

$$2(a_1 + a_2 + \cdots + a_n) = b_1 + b_2 + \cdots + b_n$$

holds for some positive integer $n$, is ______

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If

$$AP(1;3)\cap AP(2;5)\cap AP(3;7) = AP(a;d)$$

then $a + d$ equals ______

Which of the following is the only CORRECT combination?

Which of the following is the only CORRECT combination?

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1, 6, 11, \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9, 16, 23, \ldots$. Then, the number of elements in the set $X \cup Y$ is _____.

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

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

If the sum of the first n terms of the series $\sqrt3+\sqrt{75}+\sqrt{243}+\sqrt{507}+\dots$ is $435\sqrt3$, then n equals :