Answer
n-th term:$\quad a_{n}=n^{3}+1$
The next term is $\quad a_{5}=126$
Work Step by Step
Each term of the given sequence is close to the corresponding terms of
$1,\ 8$, $27$, $64$, $...\qquad$ (cubes of natural numbers)
The given sequence is interpreted as
$1^{3}+1$, $2^{3}+1$, $3^{3}+1$, $4^{3}+1$, $...,\ n^{3}+1$,...
We expect the next term to be $5^{3}+1=126$
