If $\alpha ,\;\beta ,\;\gamma $ are the geometric means between $ca,\;ab;\;ab,\;bc;\;bc,\;ca$ respectively where $a,\;b,\;c$ are in A.P., then ${\alpha ^2},\;{\beta ^2},\;{\gamma ^2}$ are in

Five numbers are in $A.P.$, whose sum is $25$ and product is $2520 .$ If one of these five numbers is $-\frac{1}{2},$ then the greatest number amongst them is

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $\mathrm{S}_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7$, then $S_{15}-S_5$ is equal to:

Let the digits $a, b, c$ be in $A.P.$ Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in $A.P.$ at least once. How many such numbers can be formed?

The arithmetic mean of first $n$ natural number

In an $A.P.,$ if $p^{\text {th }}$ term is $\frac{1}{q}$ and $q^{\text {th }}$ term is $\frac{1}{p},$ prove that the sum of first $p q$ terms is $\frac{1}{2}(p q+1),$ where $p \neq q$