If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be 

If the sum of the roots of the equation $a{x^2} + bx + c = 0$ be equal to the sum of the reciprocals of their squares, then $b{c^2},\;c{a^2},\;a{b^2}$ will be in

The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + ........$ to $9$ terms is

The sums of $n$ terms of two arithmatic series are in the ratio $2n + 3:6n + 5$, then the ratio of their ${13^{th}}$ terms is

A manufacturer reckons that the value of a machine, which costs him $Rs.$ $15625$ will depreciate each year by $20 \% .$ Find the estimated value at the end of $5$ years.

Let $a, b, c, d, e$ be natural numbers in an arithmetic progression such that $a+b+c+d+e$ is the cube of an integer and $b+c+d$ is square of an integer. The least possible value of the number of digits of $c$ is