If the sum of $n$ terms of an $A.P.$ is $2n^2 + 5n$,then the $n^{th}$ term will be

The sixth term of an $A.P.$ is equal to $2$. The value of the common difference $x$ of the $A.P.$ which makes the product $a_1 a_4 a_5$ least is given by:

If ${a_1}, {a_2}, \dots, {a_{n+1}}$ are in $A.P.$,then $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + \dots + \frac{1}{{{a_n}{a_{n+1}}}}$ is

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 the sum and product of the first three terms in an $A.P.$ are $33$ and $1155$,respectively,then a value of its $11^{th}$ term is

The value of $\sum_{n=0}^{\infty} \frac{(n+1)^2}{7^n}$ is -