Question

A = 1 + 2 + 2^2 + 2^3 + ... + 2^100

Answer 1

\(2A=2+2^2+2^3+...+2^{101}\)

=>\(A=2^{101}-1\)

Answer 2

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

\(\Rightarrow2A=2+2^2+...+2^{101}\)

\(\Rightarrow2A-A=\left(2+2^2+...+2^{101}\right)-\left(1+2+2^2+...+2^{100}\right)\)

\(\Rightarrow A=2^{101}-1\)

Answer 3

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

\(\Rightarrow2A=2+2^2+2^3+2^4+...+2^{101}\)

\(A=2A-A\)

    \(=\left(2+2^2+2^3+2^4+...+2^{101}\right)-\left(1+2+2^2+2^3+...+2^{100}\right)\)

    \(=2^{101}-1\)