Question

[Phạm Hải Đăng]

Giúp mình 2 câu này với !

Answer 1

[Yen Nhi]

`Answer:`

Bài 1:

\(S_1=2+2^3+2^5+...+2^{51}\)

\(\Rightarrow4S_1=2^3+2^5+2^7+...+2^{53}\)

\(\Rightarrow4S-S=\left(2^3+2^5+2^7+...+2^{53}\right)-\left(2+2^3+2^5+...+2^{51}\right)\)

\(\Rightarrow3S=2^{53}-2\)

\(\Rightarrow S=\frac{2^{53}-2}{3}\)

Bài 2:

\(S_2=\frac{1}{3}+\left(\frac{1}{3}\right)^3+\left(\frac{1}{3}\right)^5+...+\left(\frac{1}{3}\right)^{99}\)

\(\Rightarrow3^2.S_2=3^2.\left(\frac{1}{3}+\left(\frac{1}{3}\right)^3+\left(\frac{1}{3}\right)^5+...+\left(\frac{1}{3}\right)^{99}\right)\)

\(\Rightarrow9.S_2=3+\frac{1}{3}+\left(\frac{1}{3}\right)^3+\left(\frac{1}{3}\right)^5+...+\left(\frac{1}{3}\right)^{97}\)

\(\Rightarrow9.S_2-S_2=\left(3+\frac{1}{3}+\left(\frac{1}{3}\right)^3+...+\left(\frac{1}{3}\right)^{97}\right)-\left(\frac{1}{3}+\left(\frac{1}{3}\right)^3+\left(\frac{1}{3}\right)^5+...+\left(\frac{1}{3}\right)^{99}\right)\)

\(\Rightarrow8.S_2=3+\frac{1}{3}+\left(\frac{1}{3}\right)^3+...+\left(\frac{1}{3}\right)^{97}-\frac{1}{3}-\left(\frac{1}{3}\right)^3-\left(\frac{1}{3}\right)^5-...-\left(\frac{1}{3}\right)^{99}\)

\(\Rightarrow8.S^2=3-\left(\frac{1}{3}\right)^{99}\)

\(\Rightarrow8.S_2=\frac{3^{100}}{3^{99}}-\frac{1}{3^{99}}\)

\(\Rightarrow S_2=\frac{3^{100}-1}{3^{99}.8}\)