Consider an arithmetic series and a geometric series having four initial terms from the set $\{11,8,21,16,26,32,4\}$ If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to .......

If three unequal numbers $p,\;q,\;r$ are in $H.P.$ and their squares are in $A.P.$, then the ratio $p:q:r$ is

If $a,\;b,\;c$ are the positive integers, then $(a + b)(b + c)(c + a)$ is

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a $G.P.$ be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an $A.P.$ If the $5^{\text {th }}$ term of the $G.P.$ is the $\mathrm{n}^{\text {th }}$ term of the $A.P.$, then $\mathrm{n}$ is equal to

Let $b_i>1$ for $i=1,2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression ($A.P$.) with the common difference $\log _e 2$. 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 $s=a_1+a_2+\cdots+t_{65}$, then

If $a,\,b,\,c$ are three unequal numbers such that $a,\,b,\,c$ are in $A.P.$ and $b -a, c -b, a$ are in $G.P.$, then $a : b : c$ is