Arithmetic & Geometric Progression Questions with Solutions

Sum of squares of odd positive integers from 1 to 100 belongs to

The roots of the quadratic equation $3x^2-px+q=0$ are the ${10}^{th}$ and ${11}^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q-2p$ is:

Let $A_k$ be the arithmetic mean of the squares of the first $k$ natural numbers. If ${\sum }_{k=1}^n\left(6A_k-3k\right)=31$, find the value of $n$.

The sum of infinitely many terms of a decreasing GP is equal to the greatest value of the function $f\left(x\right)=x^3+3x-9$ in the interval $\left[-2,3\right]$, and the difference between the first two terms is $f\prime \left(0\right)$. Then the common ratio of the GP is

If a, b, c, d are in HP and the arithmetic mean of ab, bc, cd is 9, then which of the following is the value of ad?

In a G.P. consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of the G.P. is

The sum of n terms of an arithmetic series is 216. The value of the first term is n and the value of the $n^{th}$ term is 2n. The common difference d is

If $a,b,c$ are positive, $c>a$, and $a,b,c$ are in H.P., then $log\left(a+c\right)+log\left(a-2b+c\right)$ is equal to

Find $k$ in the equation $x^3-6x^2+kx+64=0$ if its roots are in geometric progression.

If $A_1=3$, $A_2=5,7,9$, $A_3=11,13,15,17,19$, $A_4=21,23,25,27,29,31,33$ and so on, what is the average of the numbers of the set $A_{20}$?

In a harmonic progression, the $p^{th}$ term is $q$ and the $q^{th}$ term is $p$. Then the ${\left(pq\right)}^{th}$ term is

If $a$, $b$, $c$ are in geometric progression, then ${log}_{ax}a$, ${log}_{bx}a$ and ${log}_{cx}a$ are in

The value of the sum $\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+\cdots +\frac{1}{25\sqrt{24}+24\sqrt{25}}$ is

The harmonic mean of two numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation $2A+G^2=27$. Then the two numbers are

Three positive numbers whose sum is 21 are in arithmetic progression. If 2, 2, 14 are added to them respectively then the resulting numbers are in geometric progression. Then which of the following is not among the three numbers?

Suppose that m and n are fixed numbers such that the mth term of an HP is equal to n and the nth term is equal to m $\left(m\ne n\right)$. Then the $\left(m+n\right)$ th term is

If $x=1+\sqrt[6]{2}+\sqrt[6]{4}+\sqrt[6]{8}+\sqrt[6]{16}+\sqrt[6]{32}$, then ${\left(1+\frac{1}{x}\right)}^{24}=$

The number of solutions of $5^{1+|sinx|+{|sinx|}^2+\cdots }=25$ for $x\in \left(-\pi ,\pi \right)$ is

If one AM (arithmetic mean) $a$ and two GMs (geometric means) $p$ and $q$ are inserted between any two positive numbers, then the value of $p^3+q^3$ is

The value of the series $\frac{2}{3!}+\frac{4}{5!}+\frac{6}{7!}+\cdots$ is