Question

[nguyen thi ca]

chứng minh:D=7¹+7²+7³+7⁴+...+7²⁰¹⁰  chia hết cho 57

Answer 1

[Minh Triều]

D=7¹+7²+7³+7⁴+...+7²⁰¹⁰

=(7¹+7²+7³)+(7⁴+7⁵+7⁶)+...(7²⁰⁰⁸+7²⁰⁰⁹+7²⁰¹⁰)

=(7.1+7.7+7.7²)+(7⁴.1+7⁴.7+7⁴.7²)+...+(7²⁰⁰⁸.1+7²⁰⁰⁸.7+7²⁰⁰⁸.7²)

=7.(1+7+7²)+7⁴.(1+7+7²)+...+7²⁰⁰⁸.(1+7+7²)

=7.57+7⁴.57+7²⁰⁰⁸.57

=57.(7+7⁴+...+7²⁰⁰⁸)

=>D chia hết cho 57

Answer 2

[Hồ Thu Giang]

D = 7+7²+7³+.....+7²⁰⁰⁰

D = (7+7²+7³)+(7⁴+7⁵+7⁶)+....+(7¹⁹⁹⁸+7¹⁹⁹⁹+7²⁰⁰⁰)

D = 7(1+7+7²)+7⁴(1+7+7²)+....+7¹⁹⁹⁸(1+7+7²)

D =7.57 + 7⁴.57 +......+ 7¹⁹⁹⁸.57

D = 57.(7+7⁴+....+7¹⁹⁹⁸) chia hết cho 57 (đpcm)