Question

2+2²+2³+...2n=2⁵⁵-2​

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