Question

A=1+7+7²+7³+...+7⁹⁹

B=1-4+4²-4³+...+4⁵⁶-4⁵⁷

Answer 1

Lời giải:

\(A=1+7+7^2+7^3+...+7^{99}\)

\(\Rightarrow 7A=7+7^2+7^3+...+7^{100}\)

Trừ vế 2 cho vế 1:

\(7A-A=(7+7^2+7^3+...+7^{100})-(1+7+7^2+..+7^{99})\)

\(\Leftrightarrow 6A=7^{100}-1\Rightarrow A=\frac{7^{100}-1}{6}\)

Xét biểu thức B

\(B=1-4+4^2-4^3+...+4^{56}-4^{57}\)

\(\Rightarrow 4B=4-4^2+4^3-4^4+...+4^{57}-4^{58}\)

Lấy vế 2 cộng vế 1:
\(B+4B=(1-4+4^2-4^3+..+4^{56}-4^{57})+(4-4^2+4^3-4^4+...+4^{57}-4^{58})\)

\(5B=1-4^{58}\Rightarrow B=\frac{1-4^{58}}{5}\)