Question

Prove by induction that 9^(n+2)- 4n is divisible by 5 for n greater than or equal to 1.

Thanks.

Answer 1

1. test for n = 1
expression is 9^(3) - 4 = 725 which is divisible by 5
2. assume it is true for n = k
that is, assume that 9^(k+2) - 4k is divisible by 5

3. show that 9^(k+1 + 2) - 4(k+1) is divisible by 5
or 9^(k+3) - 4k - 4

Using the fact that if the difference between two numbers is divisble by c and one of the numbers is divisible by c, then the other one must be also

e.g. 3423 - 21 = 3402
both 3402 21 divide by 7, so 3423 must divide by 7

So let's take the difference:
9^(k+3) - 4k - 4 - (9^(k+2) - 4k)
= 9^(k+2) [9^1 - 4]
= (5)(9^(k+2)
which is a multiple of 5, so using the above property,
9^(k+3) - 4k - 4 is also divisible by 5

Q.E.D.