Question

Cho S = 2^1 + 2^2 + ........... + 2^ 100

CMR 

S chia hết cho 3                                                      S chia hết cho 15

Answer 1

S = 2 + 2² + 2³ + ..... + 2¹⁰⁰

   =(2 + 2²) + (2³ + 2⁴) + .....+ (2⁹⁹ + 2¹⁰⁰)

   = 2.(1 + 2) + 2³.(1 + 2) + ..... + 2⁹⁹.(1 + 2)

  = 2.3 + 2³ .3 + ........ + 2⁹⁹ .3

  = 3.(2 + 2² + .... + 2⁹⁹) cia hết cho 3(đpcm)

Answer 2

S = 2 + 2² + 2³ + ..... + 2¹⁰⁰

   = ( 2 + 2² ) + ( 2³ + 2⁴ ) + .....+ ( 2⁹⁹ + 2¹⁰⁰ )

   = 2 . ( 1 + 2 ) + 2³ . ( 1 + 2 ) + ..... +  2⁹⁹ . ( 1 + 2 )

  = 2 . 3 + 2³ . 3 + ........ + 2⁹⁹ . 3

  = 3 . ( 2 + 2² + .... + 2⁹⁹ ) chia hết cho 3 => ( đpcm )

Answer 3

S = 2 + 2² + 2³ + ..... + 2¹⁰⁰

   =(2 + 2²) + (2³ + 2⁴) + .....+ (2⁹⁹ + 2¹⁰⁰)

   = 2.(1 + 2) + 2³.(1 + 2) + ..... + 2⁹⁹.(1 + 2)

  = 2.3 + 2³ .3 + ........ + 2⁹⁹ .3

  = 3.(2 + 2² + .... + 2⁹⁹) cia hết cho 3(đpcm)

Answer 4

S = 2 + 2² + 2³ + ..... + 2¹⁰⁰

   = ( 2 + 2² ) + ( 2³ + 2⁴ ) + .....+ ( 2⁹⁹ + 2¹⁰⁰ )

   = 2 . ( 1 + 2 ) + 2³ . ( 1 + 2 ) + ..... +  2⁹⁹ . ( 1 + 2 )

  = 2 . 3 + 2³ . 3 + ........ + 2⁹⁹ . 3

  = 3 . ( 2 + 2² + .... + 2⁹⁹ ) chia hết cho 3 => ( đpcm )