Arithmetic progression

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

Let $b_i>1$ for $i=1,2,\ldots,101$. Suppose $\log_eb_1,\log_eb_2,\ldots,\log_eb_{101}$ are in Arithmetic Progression (A.P.) with the common difference $\log_e2$. Suppose $a_1,a_2,\ldots,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\cdots+b_{51}$ and $s=a_1+a_2+\cdots+a_{51}$, then

If the 2$^{\text{nd}}$, 5$^{\text{th}}$ and 9$^{\text{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

Let $a_1, a_2, a_3, \dots, a_n, \dots$ be in A.P. If $a_3+a_7+a_{11}+a_{15}=72$, then the sum of its first 17 terms is equal to :

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6 : 11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ($l,n>1$) and $G_1$, $G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2G_2^4+G_3^4$ equals.

Let the sum of the first three terms of an A.P. be 39 and the sum of its last four terms be 178. If the first term of this A.P. is 10, then the median of the A.P. is :

Let $a,b,c$ be positive integers such that $\dfrac ba$ 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

The probability that $x_1$, $x_2$, $x_3$ are in an arithmetic progression, is

Let $\alpha$ and $\beta$ be the roots of equation $px^2+qx+r=0$, $p\ne0$. If $p$, $q$, $r$ are in A.P. and $\dfrac1\alpha+\dfrac1\beta=4$, then the value of $|\alpha-\beta|$ is :

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is :

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4$^{\text{th}}$ term is :

The sum of the first 20 terms common between the series $3+7+11+15+\dots$ and $1+6+11+16+\dots$, is :

The number of terms in an A.P. is even; the sum of the odd terms in it is 24 and that the even terms is 30. If the last term exceeds the first term by $10\frac12$, then the number of terms in the A.P. is :

A pack contains $n$ cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is $k$, then $k-20=$

Let $a_1,a_2,a_3,\ldots,a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\displaystyle\sum_{i=1}^pa_i$, $1\le p\le100$. For any integer $n$ with $1\le n\le20$, let $m=5n$. If $\dfrac{S_m}{S_n}$ does not depend on $n$, then $a_2$ is

Let $a_1,a_2,a_3,\ldots,a_{11}$ be real numbers satisfying $$a_1=15,\quad 27-2a_2>0\ \text{ and }\ a_k=2a_{k-1}-a_{k-2}\ \text{ for } k=3,4,\ldots,11.$$ If $\dfrac{a_1^2+a_2^2+\cdots+a_{11}^2}{11}=90$, then the value of $\dfrac{a_1+a_2+\cdots+a_{11}}{11}$ is equal to

If the sum of first $n$ terms of an A.P. is $cn^2$, then the sum of squares of these $n$ terms is

The sum $V_1+V_2+\cdots+V_n$ is

$T_r$ is always