Question

if n belongs to N, then 11^n+2+12^2n+1​ is divisible by

Answer 1

Step-by-step explanation:

ANSWER

First, prove it for the base case ofn=1:

11

(1+2)

+12

(2+1)

= 11

3

+12

3

= 1331+1728

= 133(10)+1+1728

=133(10)+1729

=133(10)+133(13)

=133(26)

Assume it is true for a natural number k. Prove it is true for the number k+1:

True for k:

11

(k+2)

+12

(2k+1)

=133(m), where m is an integer.

Fork+1:

11

k+1+2

+12

2(k+1)+1

= 11

k+2

×11+12

2k+2+1

=11

(

k+2)×11+12

(

2k+1)×12

2

=11×11

k+2

+(11+133)×12

2k+1

=11(11

k+2

+12

2k+1

)+133×12

2k+1

=11(133m)+133×12

2k+1

= 133(11m+12

(2k+1)

) hence it is divisible by 133.