Let the sequence $a_{n}$ be defined as follows:

${a_1} = 1,{a_n} = {a_{n – 1}} + 2$ for $n\, \ge \,2$

Find first five terms and write corresponding series.

Suppose the sum of the first $m$ terms of an arithmetic progression is $n$ and the sum of its first $n$ terms is $m$, where $m \neq n$. Then, the sum of the first $(m+n)$ terms of the arithmetic progression is

Let ${\left( {1 – 2x + 3{x^2}} \right)^{10x}}  = {a_0} + {a_1}x + {a_2}{x^2} + …..+{a_n}{x^n},{a_n} \ne 0$, then the arithmetic mean of $a_0,a_1,a_2,…a_n$ is

If  ${\log _5}2,\,{\log _5}({2^x} – 3)$ and  ${\log _5}(\frac{{17}}{2} + {2^{x – 1}})$ are in $A.P.$ then the value of $x$ is :-

The number of terms in an $A .P.$ is even ; the sum of the odd terms in it is $24$ and that the even terms is $30$. If the last term exceeds the first term by $10\frac{1}{2}$ , then the number of terms in the $A.P.$ is