Question

Write the first five terms and obtain the corresponding series, whose sequence is $a_1=a_2=2,a_n=a_{n-1}-1,n>2$.

Answer 1

Finding sequence and corresponding series.
Given, $a_1=a_2=2$ and $a_n=a_{n-1}-1\forall n>2,n\in N$
Substituting $n=3,4$ and $5$ in $a_n$, we get

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

$a_4=a_{4-1}-1$
$\Rightarrow a_4=a_3-1$
$\Rightarrow a_4=1-1$
$\Rightarrow a_4=0$

$a_5=a_{5-1}-1$
$\Rightarrow a_5=a_4-1$
$\Rightarrow a_5=0-1$
$\Rightarrow a_5=-1$

Hence, the first five terms of the sequence are $2,2,1,0,-1$ and corresponding series is $2+2+1+0+(-1)+\cdots$.