What is the formula of: $a^{0} + a^{1} + a^{2} + ... + a^{n-1} + a^{n}$? What is the formula of:
$$a^{0} + a^{1} + a^{2} + ... +  a^{n-1} + a^{n}$$
Any ideas?

If $a=1,a^{0} + a^{1} + a^{2} + ... +  a^{n-1} + a^{n}=1+1+\cdots$ up to $(n+1)$  terms hence $=n+1$
Else let $S=a^{0} + a^{1} + a^{2} + ... +  a^{n-1} + a^{n}$
So, $a\cdot S=a^{1} + a^{2} + a^{3} + ... +  a^{n} + a^{n+1}$
So, $S(a-1)=a^{n+1}-1$
