Question

Cho S = 2^1+2^2+2^3+...+2^100. Chứng minh S chia hết cho 15

Answer 1

A=2+2^2+2^3+...+2^100

  = (2+2^2+2^3+2^4)+...(2^97+2^98+2^99+2^100)

  =2(1+2+2^2+2^3)+....+2^97(1+2+2^2+2^3)

  = 2.15 +.....+2^97.15

  =(2+....+2^97).15 chia hết cho 15

Answer 2

S = 2¹ + 2² + 2³ + 2⁴ + .... + 2¹⁰⁰

S = ( 2¹ + 2² + 2³ + 2⁴ + .... + ( 2⁹⁷ + 2⁹⁸ + 2⁹⁹ + 2¹⁰⁰ )

S = 2 . ( 1 + 2 + 4 + 8 ) +.... + 2⁹⁷ . ( 1 + 2 + 4 + 8 )

S = 2 . 15 + ... + 2⁹⁷ . 15

S = ( 2 + ... + 2⁹⁷ ) . 15

Mà 15 chia hết cho 15 suy ra S chia hết cho 15