Question

the sequence {an} is defined by a1 = 2 and an+1=an+2n what is the value of a100

Answer 1

Given, $a_1=2$
$a_{n+1}=a_n+2n$

we know, $a_n=a_1+(n-1)d$ .....(i)
where d is the common difference of sequence.

now, $a_{n+1}=a_1+(n+1-1)d$
$a_{n+1}=a_1+nd$.......(ii)

put eqs. (i) and (ii) in $a_{n+1}=a_n+2n$
so, $a_1+nd=a_1+(n-1)d+2n$
nd = (n - 1)d + 2n
d = 2n .......(iii)

now, $a_{100}=a_1+(100-1)d$
= 2 + 99d
= 2 + 99 × 2n [ from eq. (iii)
= 2 + 198n