Question

Chứng minh  A=75.(4\(^{1975}\)+4\(^{1974}\)+...+4\(^2\)+5) +25 chia het cho 4\(^{1976}\)

Answer 1

Đặt    S=4¹⁹⁷⁵+4¹⁹⁷⁴+...+4²

=>   4S=4¹⁹⁷⁶+4¹⁹⁷⁵+...+4³

=>4S-S=4¹⁹⁷⁶+4¹⁹⁷⁵+...+4³-4¹⁹⁷⁵-4¹⁹⁷⁴-...-4²

=>    3S=4¹⁹⁷⁶-4²

^{=>       \(S=\frac{4^{1976}-16}{3}\)}

=>   \(A=75.\left(4^{1975}+4^{1974}+...+4^2+5\right)+25\)

=>   \(A=75.\left(S+5\right)+25\)

=>   \(A=75.\left(\frac{4^{1976}-16}{3}+\frac{15}{3}\right)+25\)

=>   \(A=75.\frac{4^{1976}-1}{3}+25\)

=>   \(A=25.\left(4^{1976}-1\right)+25\)

=>   \(A=25.4^{1976}-25+25\)

=>   \(A=25.4^{1976}\)

=>  

A chia hết cho 4¹⁹⁷⁶

=> ĐPCM

Answer 2

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