Prove by induction that $(x+1)^n – nx – 1$ is divisible by $x^2$
Basis step has already been completed. I've started off with the inductive step as just $n=k+1$, trying to involve $f(k)$ into it so that the left over parts can be deducible to be divisible by $x^2$ but getting stuck on this inductive step.
Best Answer
By the Binomial theorem, $(x+1)^n$ ends in $nx+1$, so that the remaining terms are multiples of $x^2$.
By induction, $(x+1)^n$ ends in $nx+1$. This is true for $n=1$, $(x+1)^1$ ends in $x+1$.
Now assume that $(x+1)^n$ ends in $nx+1$. Multiplying by $x+1$, we get $nx+x+1$ and higher order terms, hence $(x+1)^{n+1}$ ends in $(n+1)x+1$.