Using induction prove 8 divides 7n 3n 2SolutionPn7n 3n 2

Using induction, prove 8 divides 7^n + 3^n - 2

Solution

P(n):7^n + 3^n - 2

put n=1

P(1)=7¹ +3¹ -2=7+3-2 =8

8 dvisble by 8

so P(1) is true

Assume p(K) is true

P(k)=7^k+ 3^k - 2

P(k+1)=7^(k+1)+3(k+1)-2

=7^k .7+3^{k .}.3-2

=7(7^k +3^k) -4 3^k +14

=7(7^k +3^k -2]+14 +4.3^k -2

=4[3^k +3]

[3^k +3] is always even and 3^k is always odd

so 3^k +3 is always divisibe by 2

4*2x --->8

divisible by 8

hence

P(k+1) is true

by principle of mathematical induction

for P(n )belongs to N

8 divides 7^n + 3^n - 2