Question

Tính:

A = 1/2 - 1/2² + 1/2³ - 1/2⁴ + ... + 1/2⁹⁹ - 1/2¹⁰⁰

Answer 1

\(A=\dfrac{1}{2}-\dfrac{1}{2^2}+\dfrac{1}{2^3}-\dfrac{1}{2^4}+...+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}\)

=>\(2A=1-\dfrac{1}{2}+\dfrac{1}{2^2}-\dfrac{1}{2^3}+...+\dfrac{1}{2^{98}}-\dfrac{1}{2^{99}}\)

=>\(2A+A=1-\dfrac{1}{2}+\dfrac{1}{2^2}-...+\dfrac{1}{2^{98}}-\dfrac{1}{2^{99}}+\dfrac{1}{2}-\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}-\dfrac{1}{2^{100}}\)

=>\(3A=1-\dfrac{1}{2^{100}}=\dfrac{2^{100}-1}{2^{100}}\)

=>\(A=\dfrac{2^{100}-1}{2^{100}\cdot3}\)