The number of terms common to the two A.P.'s $3,7,11, \ldots ., 407$ and $2,9,16, \ldots . .709$ is

If three numbers be in $G.P.$, then their logarithms will be in

The sum of $n$ terms of two arithmetic progressions are in the ratio $(3 n+8):(7 n+15) .$ Find the ratio of their $12^{\text {th }}$ terms.

If ${S_k}$ denotes the sum of first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively, then ${S_{kn}}/{S_n}$ be independent of $n$ if

If $(b+c),(c+a),(a+b)$ are in $H.P$ , then $a^2,b^2,c^2$ are in.......

If the variance of the terms in an increasing $A.P.$, $b _{1}, b _{2}, b _{3}, \ldots b _{11}$ is $90,$ then the common difference of this $A.P.$ is