If $m$ arithmetic means are inserted between $1$ and $31$ such that the ratio of the $7^{th}$ mean to the $(m - 1)^{th}$ mean is $5:9$,then the value of $m$ is:

For three positive integers $p, q, r$,$x^{pq p^2} = y^{qr} = z^{p^2 r}$ and $r = pq + 1$ such that $3, 3 \log_y x, 3 \log_z y, 7 \log_x z$ are in $A$.$P$. with common difference $\frac{1}{2}$. Then $r - p - q$ is equal to

Let $9 < x_1 < x_2 < \ldots < x_7$ be in an $A.P.$ with common difference $d$. If the standard deviation of $x_1, x_2, \ldots, x_7$ is $4$ and the mean is $\overline{x}$,then $\overline{x} + x_6$ is equal to:

The common difference of the $A.P.: a_{1}, a_{2}, ..., a_{m}$ is $13$ more than the common difference of the $A.P.: b_{1}, b_{2}, ..., b_{n}$. If $b_{31} = -277$,$b_{43} = -385$ and $a_{78} = 327$,then $a_{1}$ is equal to

Four numbers are in an arithmetic progression. The sum of the first and last terms is $8$ and the product of the two middle terms is $15$. What is the smallest number in the sequence?

The first term of an $A.P.$ is $2$ and the common difference is $4$. The sum of its $40$ terms will be: