Sets

Question

Prove by mathematical induction that sum of cubes of three consecutive natural numbers is divisible by 9.

Answer

Let n, n+1, n+2 be three consecutive natural numbers.

Let P(n): is divisible by 9.**I. **For n = 1,

is divisible by 9

1 + 8 + 27 is divisible by 9 36 is divisible by 9

which is true

∴ the statement is true for n = 1.**II. **Suppose the statement is true for n = m,

P(m) : is divisible by 9.

...(i)**III. **For n = m + 1,

is divisible by 9.

Now, from (i),

where

is divisible by 9

P (m + 1) is true

∴ P (m) is true P (m + 1) is true.

Hence, by the principal of mathematical induction, P (n) is true for all