If the sum of the first $2n$ terms of $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of $57, 59, 61, \dots$,then $n$ is equal to

If the sum of $n$ terms of an $A$.$P$. is $3n^2 + 5n$ and its $m$th term is $164$, then the value of $m$ is

Let $a_1, a_2, a_3, \ldots, a_{11}$ be real numbers satisfying $a_1=15$,$27-2a_2 > 0$,and $a_k = 2a_{k-1} - a_{k-2}$ for $k = 3, 4, \ldots, 11$. If $\frac{a_1^2 + a_2^2 + \ldots + a_{11}^2}{11} = 90$,then the value of $\frac{a_1 + a_2 + \ldots + a_{11}}{11}$ is equal to

If $a_1, a_2, a_3, \dots, a_n$ are in an arithmetic progression with common difference $d$,then find the value of $\sin d [\csc a_1 \csc a_2 + \csc a_2 \csc a_3 + \dots + \csc a_{n-1} \csc a_n]$.

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$

If the sum of the first four terms of an $A.P.$ is $6$ and the sum of its first six terms is $4$, then the sum of its first twelve terms is