Question

Thu gọn
S = \(\dfrac{1}{4}+\dfrac{1}{4^2}+\dfrac{1}{4^3}+\dfrac{1}{4^4}+\dfrac{1}{4^5}+\dfrac{1}{4^6}+...+\dfrac{1}{4^{29}}+\dfrac{1}{4^{30}}\)

Answer 1

\(S=\dfrac{1}{4}+\dfrac{1}{4^2}+\dfrac{1}{4^3}+...+\dfrac{1}{4^{30}}\)

\(\Rightarrow4S=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{29}}\)

\(\Rightarrow3S=4S-S=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{29}}-\dfrac{1}{4}-\dfrac{1}{4^2}-...-\dfrac{1}{4^{30}}=1-\dfrac{1}{4^{30}}\)

\(\Rightarrow S=\dfrac{1-\dfrac{1}{4^{30}}}{3}\)