Question

[Mạc Trần Sang]

chứng minh S = 5 + 5^2 + 5^3 + ... + 5^96 chia hết cho 126

giúp mk nhé, cảm ơn nhiều

Answer 1

[Nguyệt]

\(S=\left(5+5^4\right)+\left(5^2+5^5\right)+...+\left(5^{93}+5^{96}\right)\)

\(S=5.\left(1+5^3\right)+5^2.\left(1+5^3\right)+...+5^{93}.\left(1+5^3\right)\)

\(S=5.125+5^2.125+...+5^{93}.125\)

\(S=125.\left(5+5^2+...+5^{93}\right)⋮125\)

Answer 2

[Ad]

\(S=5+5^2+5^3+...+5^{96}\)(có 96 số, 96 chia hết cho 6)

\(=\left(5+5^2+5^3+5^4+5^5+5^6\right)+...+\left(5^{91}+5^{92}+5^{93}+5^{94}+5^{95}+5^{96}\right)\)

\(=\left(5+5^4\right)+\left(5^2+5^5\right)+\left(5^3+5^6\right)+...+\left(5^{91}+5^{94}\right)+\left(5^{92}+5^{95}\right)+\left(5^{93}+5^{96}\right)\)

\(=5.\left(1+5^3\right)+5^2.\left(1+5^3\right)+...+5^{92}.\left(1+5^3\right)+5^{93}.\left(1+5^3\right)\)

\(=5.126+5^2.126+5^3.126+...5^{91}.126+5^{92}.126+5^{93}.126\)

\(=126.\left(5+5^2+5^3+...+5^{91}+5^{92}+5^{93}\right)\)chia hết cho 126.

Vậy \(S=5+5^2+5^3+...+5^{96}\)chia hết cho 126.