Question

Rút gọn A= 1/2 + 1/2^2 +......+1/2^100

B= 1/3 + 1/3^2 +............+ 1/3^100

Answer 1

A= \(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{100}}\)

2A= \(2.\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{100}}\right)\)

2A= \(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}\)

⇒ 2A- A= \(1-\dfrac{1}{2^{100}}\)

⇒ A= \(1-\dfrac{1}{2^{100}}\)

B= \(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{100}}\)

3B= \(3.\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{100}}\right)\)

3B= \(1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{99}}\)

⇒ 3B- B= \(1-\dfrac{1}{3^{100}}\)

⇒ B.(3-1)= \(1-\dfrac{1}{3^{100}}\)

⇒ 2B= \(1-\dfrac{1}{3^{99}}\)

⇒ B= \(\left(1-\dfrac{1}{3^{99}}\right):2\)

⇒ B= \(\dfrac{1}{2}-\dfrac{1}{2.3^{99}}\)