Question

Aflati nr natural n care verifica egalitatile:  2^1+2^2+2^3+....2^n =2^55 - 2

Answer 1

2¹+2²+2³+...+2^n=2⁵⁵-2
Daca inmultim toata propozitia cu 2, avem:

2²+2³+2⁴+...+2^n+2^(n+1)=2⁵⁶-2² 
Adaugam (si scadem) un 2:

2   -2+   2²+2³+2⁴+...+2^n+2^(n+1)=2⁵⁶-2² 
(2 +2²+2³+2⁴+...+2^n)  +  2^(n+1) -2= 2⁵⁶-2² 
2⁵⁵-2 +  2^(n+1) -2= 2⁵⁶-2² 
2⁵⁵ +  2^(n+1) -4= 2⁵⁶-4 
2⁵⁵ +  2^(n+1) = 2⁵⁶
2^(n+1) = 2⁵⁶-2⁵⁵
2^(n+1) = 2⁵⁵(2-1)
2^(n+1) = 2⁵⁵
2^(n+1) = 2⁵⁴⁺¹
n+1=54+1
n=54