Question

A=4+4²+4³+...+4¹⁹+4²⁰

B=1+3+3²+3³+...3⁹⁹

Answer 1

\(A=4+4^2+4^3+...+4^{19}+4^{20}\)

\(4A=4\cdot\left(4+4^2+4^3+...+4^{20}\right)\)

\(4A=4^2+4^3+...+4^{21}\)

\(4A-A=4^2+4^3+4^4+...+4^{21}-4-4^2-4^3-...-4^{20}\)

\(3A=4^{21}-4\)

\(A=\dfrac{4^{21}-4}{3}\)

____________

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

\(3B=3\cdot\left(1+3+3^2+....+3^{99}\right)\)

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

\(3B-B=\left(3+3^2+3^3+....+3^{100}\right)-\left(1+3+3^2+...+3^{99}\right)\)

\(2B=3^{100}-1\)

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

Answer 2

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