Properties of an Arithmetic Progression

If $\frac{1}{p+q}, \frac{1}{q+r} \mathrm{and} \frac{1}{r+p}$ are in AP, then 

If $a^2\left(b+c\right), b^2\left(c+a\right)$ and $c^2\left(a+b\right)$ are in A.P., then $a, b, c$ are in

If $p,q,r$ re in H.P. and $p,q,-2r$ are in G.P. $\left(p,q,r>0\right)$ then

Find the three numbers $a,b,c$ between $2$ and $18$ such that

(i) their sum is $25$

(ii) the numbers $2,a,b$ are consecutive terms of an AP

(iii) the numbers $b,c,18$ are consecutive terms of GP

If $\frac{1}{q+r},\frac{1}{r+p},\frac{1}{p+q}$ are in A.P., then 

Let $a_1,a_2,a_3\dots ,a_n$ be in A.P. and $a_3,a_5,a_8,b_1,b_2,b_3\dots ,b_n$ be in G.P. If $a_9=40$, then the value of $\sum _{i=1}^2{a}_i^2$ is

Six numbers are in A.P. such that their sum is $3$. The first term is $4$ times the third term. Then the fifth term is

In a triangle $ABC$, if $\tan \left(\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right),\tan \left(\raisebox{1ex}{$B$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right) \text{and} \tan \left(\raisebox{1ex}{$C$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ are in Arithmetic Progression, then which of the following option is always correct$?$

${\log }_{10}2, {\log }_{10}\left(2^x-1\right)$ and ${\log }_{10}\left(2^x+3\right)$ are three consecutive terms of an $\mathrm{AP}$ for

If $a_1, a_2,a_3\dots .$ are terms of $AP$ such that $a_1+a_5+a_{10}+a_{15}+a_{20}+a_{24}=225,$ then the sum of first $24$ terms is

If $\log 2, \log \left(2^n-1\right)$ and $\log \left(2^n+3\right)$ are in arithmetic progression, then the value of $n$ will be

If $\log 2, \log \left(2^n-1\right)$ and $\log \left(2^n+3\right)$ are in arithmetic progression, then the value of $n$ will be

If $\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{a}+\sqrt{b}}$ are in A.P. then $9^{ax+1},9^{bx+1},9^{cx+1},x\ne 0$ are in

For any three positive real numbers $a, b$ and $\mathrm{c}$ if $4\left(a^2+9b^2\right)+3\left(3c^2\mathit{-}2ac\right)=6b(2a+3c)$, then

If the sum of three numbers of a arithmetic sequence is $15$ and the sum of their squares is $83$, then the numbers are

Three numbers are in A.P. whose sum is $33$ and product is $792$, then the smallest number from these numbers is
 

if $1, {\log }_9\left(3^{1-x}+2\right), {\log }_3\left(4·3^x-1\right)$ are in $\mathrm{A}.\mathrm{P}.$, then $x$ equals

If $\log 2, \log \left(2^n-1\right)$ and $\log \left(2^n+3\right)$ are in $\mathrm{A}.\mathrm{P}.$, then $n=$

If $a, b, c$ are in A.P., then $\frac{a}{bc}, \frac{1}{c}, \frac{1}{b}$ are in