Sets

Question

Prove the following by using the principle of mathematical induction for all

n (n + 1) (n + 5) is a multiple of 3.

Answer

Let P(n): n (n + 1) (n + 5) is a multiple of 3.**I. **For n = 1,

is a multiple of 3

1 (2) (6) is a multiple of 3 12 is a multiple of 3.

which is true.

∴ P(n) is true for n = 1.**II. **Suppose P (n) is true for n = m

P(m) : m(m + 1) (m + 5) is a multiple of 3 m (m + 1) (m + 5) = 3k

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

P(m + 1): (m + 1) (m + 1 + 1) (m + 1 + 5) is a multiple of 3.

(m + 1) (m + 2) (m + 6) is a multiple of 3.

Now, (m + 1) (m + 2) (m + 6) = (m + 1)() =

= [BY (i)]

=

where

(m + 1) (m + 2) (m + 6) is a multiple of 3.

∴ P(m + 1) is true.

∴ P(m) is true P(m + 1) is true.

Hence, P(n) is true for all