If $a, b, c$ are in $GP$ and $4a, 5b, 4c$ are in $AP$ such that $a + b + c = 70$,then the value of $a^3 + b^3 + c^3$ is

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. Then the greatest value of $n$ for which $a_n$ is the greatest is

Define a sequence $\langle a_n \rangle$ by $a_1 = 5, a_n = a_1 a_2 \dots a_{n-1} + 4$ for $n > 1$. Then,$\lim_{n \to \infty} \frac{\sqrt{a_n}}{a_{n-1}}$

Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $b_{1}, b_{2}, \ldots, b_{10}$ be a $GP$ with common ratio $2$. Let $c_{k}=a_{k}+b_{k}, k=1, 2, \ldots, 10$. If $c_{2}=12$ and $c_{3}=13$,then $\sum_{k=1}^{10} c_{k}$ is equal to:

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

If all the terms of an $A.P.$ are squared,then the new series will be in