Question

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.

Answer 1

Finding sequence
Given, $a=1$ and $a_n=a_{n-1}+2$ for $n\ge 2,n\in N$

Substituting $n=2,3,4$ and $5$ in $a_n$, we get
$a_2=a_{2-1}+2$
$\Rightarrow a_2=a_1+2$
$\Rightarrow a_2=1+2$
$\Rightarrow a_2=3$

$a_3=a_{3-1}+2$
$\Rightarrow a_3=a_2+2$
$\Rightarrow a_3=3+2$
$\Rightarrow a_3=5$

$a_4=a_{4-1}+2$
$\Rightarrow a_4=a_3+2$
$\Rightarrow a_4=5+2$
$\Rightarrow a_4=7$

$a_5=a_{5-1}+2$
$\Rightarrow a_5=a_4+2$
$\Rightarrow a_5=7+2$
$\Rightarrow a_5=9$

Hence, the first five terms of the sequence are $1,3,5,7$ and $9$.
The corresponding series is
$1+3+5+7+9+\cdots$.