Question

Sa se deterine numarul natural x din:
$2+2^{2}+ 2^{3}+ 2^{4}+...+ 2^{x}= 2^{55}-2$
(clasa a 6 a)

Answer 1

   
[tex]\text{Folosim formula: } \\ 2^0+2^1+2^2+2^3+2^4+\cdots+2^n=2^{n+1} -1\\ \text{Echivalenta cu:}\\ 1+2+2^2+2^3+2^4+\cdots+2^n=2^{n+1}-1\\\\ \text{In enuntul tau nu avem primul termen (1) din formula generala.}\\ \text{Din acest motiv, rezultatul va fi cu 1 mai mic.}\\\\ \text{Rezolvare:}\\\\ 2+2^{2}+2^{3}+2^{4}+...+2^{x}=2^{55}-2\\\\ 2+2^{2}+2^{3}+2^{4}+...+2^{x}=2^{x+1}-1-1\\\\ 2^{x+1}-1-1= 2^{55}-2\\\\ 2^{x+1}-2=2^{55}-2\\ 2^{x+1}=2^{55}\\ x+1=55 \\ x=55-1=\boxed{54} [/tex]