Question

Using Mathematical induction, prove that 10n

+ 3·4n+2 + 5 is divisible by 9 for an nN.​

Answer 1

Step-by-step explanation:

10

n

+3.4

n+2

+5

Put n=1

10+3(64)+5=207 which is divisible by 3,9,23,207

We will check for n=2

100+768+5=873

which is divisible by 3 and 9 only.

Every number divisible by 9 is also divisible by 3.

Now, let P(n) is true

i.e. 10

n

+3.4

n+2

+5 is divisible by 9

⇒10

n

+3.4

n+2

+5=9λ ....(1)

Now, we will check for P(n+1)

Consider, 10

n+1

+3.4

n+3

+5

=10

n

10+3.4

n+2

4+5

=10(9λ−3.4

n+2

−5)+3.4

n+2

4+5 (by (1))

=90λ−27.4

n+2

−45

=9(10λ−3.4

n+2

−5)