If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

For every natural number $n$,$n(n + 1)$ is always

If $\frac{b + a}{b - a} = \frac{b + c}{b - c}$,then $a, b, c$ are in

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:

If $t_n$ is the $n^{th}$ term of an arithmetic progression and $t_7 = 9$,what is the value of the common difference $d$ that minimizes the product $t_1 t_2 t_7$?

Two sequences $\{t_n\}$ and $\{s_n\}$ are defined by $t_n = \log \left( \frac{5^{n+1}}{3^{n-1}} \right)$ and $s_n = \left[ \log \left( \frac{5}{3} \right) \right]^n$. Then: