Question

1.so sánh

a) A =1+2+2^2+...+2^63

    B=2^64-1

b) A= 3^4+3^5+...+3^20

    B=(3^21-3^4):2

 

 

 

Answer 1

a) \(A=1+2+2^2+...+2^{63}\)

\(\Rightarrow2A=2.\left(1+2+2^2+...+2^{63}\right)\)

\(\Rightarrow2A=2+2^2+...+2^{64}\)

\(\Rightarrow2A-A=2+2^2+...+2^{64}-\left(1+2+2^2+...+2^{63}\right)\)

\(\Rightarrow A=2+2^2+...+2^{64}-1-2-2^2-...-2^{63}\)

\(\Rightarrow A=2^{64}-1\)

Vì \(2^{64}-1=2^{64}-1\Rightarrow A=B\)

b) \(A=3^4+3^5+...+3^{20}\)

\(\Rightarrow3A=3^5+3^6+...+3^{21}\)

\(\Rightarrow3A-A=3^5+3^6+...+3^{21}-3^4-3^5-...-3^{20}\)

\(\Rightarrow2A=3^{21}-3^4\)

\(\Rightarrow A=\frac{3^{21}-3^4}{2}\)

Mà \(B=\frac{3^{21}-3^4}{2}\Rightarrow A=B\)