Question

chứng minh rằng:

S=5+5²+5³+5⁴⁺5⁵+5⁵+...+5²⁰¹⁷+5²⁰¹⁸+5²⁰¹⁹ chhia hết cho 31

khó quá mọi người giúp em với ạ

Answer 1

S= (5 +5^2+5^3) +(5^4+5^5+5^6)+...+(5^2017+5^2018+5^2019)

=5(1+5+5^2)+5^4(1+5+5^2)+...+5^2017(1+5+5^2)

=5.31+5^4.31+...+5^2017.31

=31.( 5+5^4+...+5^2017) chia hết cho 31 (đpcm)

Answer 2

S = 5 + 5² + 5³ + ... + 5²⁰¹⁷ + 5²⁰¹⁸ + 5²⁰¹⁹

S = 5 . ( 1 + 5 + 5² ) + ... + 5²⁰¹⁷ . ( 1 + 5 + 5² )

S = 5 . 31 + ... + 5²⁰¹⁷ . 31

S = 31 . ( 5 + ... + 5²⁰¹⁷ ) \(⋮\)31

Vậy : S = 5 + 5² + 5³ + ... + 5²⁰¹⁷ + 5²⁰¹⁸ + 5²⁰¹⁹ chia hết cho 31

Answer 3

Ta có : S=5+5²+5³+...+5²⁰¹⁹

             =(5+5³+5⁵)+(5²+5⁴+5⁶)+...+(5²⁰¹⁵+5²⁰¹⁷+5²⁰¹⁹)

             =5(1+5²+5⁴)+5²(1+5²+5⁴)+...+5²⁰¹⁵(1+5²+5⁴)

            =5.651+5².651+...+5²⁰¹⁵.651

Vì 651\(⋮\)31 nên 5.651+5².651+...+5²⁰¹⁵.651\(⋮\)31

hay S\(⋮\)31

Vậy S\(⋮\)31.

Answer 4

S= (5+5²+5³)+(5⁴+5⁵+5⁶)+.....+(5²⁰¹⁷+5²⁰¹⁸+5²⁰¹⁹)

S= 1(5+5²+5³)+5³(5+5²+5³)+......+5²⁰¹⁶(5+5²+5³)

S=155 .(1+5³+.......+5²⁰¹⁶)

Do 155\(⋮\)31\(\Rightarrow\) S \(⋮\) 31

Answer 5

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