Question

CMR:

P = 1 + 2 + 2² + 2³ + ... + 2¹³ + 2¹⁴ chia hết cho 31.

Answer 1

Ta có :

\(P=1+2+2^2+.........................+2^{14}\)

\(\Rightarrow P=\left(1+2+2^2+2^3+2^4\right)+........+\left(2^9+...+2^{14}\right)\)

\(\Rightarrow P=2\left(1+2+....+2^4\right)+.....+2^{10}\left(1+2+...+2^4\right)\)

\(\Rightarrow P=2.31+......+2^{10}.31\)

\(\Rightarrow P=31\left(2+...+2^{10}\right)⋮31\)

\(\rightarrowđpcm\)

Answer 2

Ta có:

P=1+2+2²+2³+...+2¹³+2¹⁴

=(1+2+2²+2³+2⁴)+(2⁵+2⁶+2⁷+2⁸+2⁹)+(2¹⁰+2¹¹+2¹²+2¹³+2¹⁴)

=31+2⁵(1+2+2²+2³+2⁴)+2¹⁰(1+2+2²+2³+2⁴)=31+2⁵.31+2¹⁰.31\(⋮\)31

Answer 3

Hình như bài này phải giải là:

P = 1 + 2 + 2² + 2³ + ... + 2¹³ + 2¹⁴

=> P = ( 1 + 2 + 2² + 2³ + 2⁴ ) + ( 2⁵ + 2⁶ + 2⁷ + 2⁸ + 2⁹ ) + ( 2¹⁰ + 2¹¹ + 2¹² + 2¹³ + 2¹⁴ )

= 31 + 2⁵. ( 1 + 2 + 2² + 2³ + 2⁴ ) + 2¹⁰. ( 1 + 2 + 2² + 2³ + 2⁴ )

= 31 + 2⁵ . 31 + 2¹⁰ . 31

= 31 . ( 1 + 2⁵ + 2¹⁰ ) chia hết cho 3.