Question

Tính tổng

a) A = 2 + 2² + 2⁵ + ...... + 2⁹⁹

b) B = 2 + 2⁴ + 2⁷ +....... + 2¹⁰⁰

c) C = 1² + 2² + 3² + ......... + 100²

d) D = 1³ + 2³ + 3³ + ........... + 100³

Answer 1

a) \(A=2+2^2+2^3+2^4+.....+2^{98}+2^{99}\)

\(\Rightarrow2A=2^2+2^3+2^4+2^5.....+2^{99}+2^{100}\)

\(\Rightarrow2A-A=\left(2^2+2^3+2^4+2^5.....+2^{99}+2^{100}\right)-\left(2+2^2+2^3+2^4+.....+2^{98}+2^{99}\right)\)

\(\Rightarrow A=2^{100}-2\)

b) \(B=2+2^4+2^7+......+2^{97}+2^{100}\)

\(\Rightarrow2^3B=2^4+2^7+......+2^{100}+2^{103}\)

\(\Rightarrow8.B-B=\left(2^4+2^7+......+2^{100}+2^{103}\right)-\left(2+2^4+2^7+......+2^{97}+2^{100}\right)\)

\(\Rightarrow7B=2^{103}-2\)

\(\Rightarrow B=\dfrac{2^{103}-2}{7}\)