Question

A= 2+$2^{2}$+$2^{3}$+.... + $2^{2010}$

Answer 1

$A=2(1+2+ 2^{2}+...+ 2^{2009})=2* (2^{2010}-1) = 2^{2011}-2$

Aici am aplicat formula: $1+a+ a^{2}+...+ a^{n}= \frac{ a^{n+1}-1 }{a-1}$, dar daca iti cere sa o demonstrezi, procedezi asa:

Fie $N=1+2+ 2^{2}+... 2^{2009} \\ 2N=2+ 2^{2} +2^{3}+...+2^{2010} \\ 2N-N=(2+ 2^{2} + 2^{3}+...+ 2^{2010})-(1+2+2^{2}+...+ 2^{2009})<=> \\ <=>N= 2^{2010}-1$