Sets

Question

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

is a multiple of 27 for all

Answer

Let P(n): is a multiple of 27.**I. **For n = 1,

P(1): is a multiple of 27

41 - 14 is a multiple of 27 27 is a multiple of 27

which is true.

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

P(m) : is a multiple of 27 ...(i)**III.** For n = m + 1,

P (m + 1) : is a multiple of 27

But,

= [By (i)]

where

is a multiple of 27

∴ P(m + 1) is true.

∴ P(m) is true P(m + 1) is true.

Hence, by induction, P(n) is true for all