Question

sa se restrângă sumele 2 plus 2 la puterea 2 plus doi la puterea 3 plus 2 la puterea 4 plus 3 la puterea 5 plus .... plus 2 la puterea 50

Answer 1

$S=2+2^2+2^3+2^4+2^5+\ldots+2^{50}.\;Calcul\u{a}m\;2S:\\\\2S=2^2+2^3+2^4+2^5+2^6+\ldots+2^{51}.\;Efectu\u{a}m\;sc\u{a}derea\;2S-S:\\\\2S-S=2^2+2^3+2^4+2^5+2^6+\ldots+2^{50}+2^{51}-(2+2^2+2^3+2^4+2^5+\ldots+2^{50})=2^{51}-2=2\cdot(2^{50}-1).\\\\Deci:\;S=2\cdot(2^{50}-1).$