Prove the statements by the Principle of Mathematical Induction :n3

1.	Prove the statements by the Principle of Mathematical Induction :n3 – 7n + 3 is divisible by 3, for all natural numbers n.
Answer» Let P(n): n³ – 7n + 3 is divisible by 3, for all natural numbers n. Now P(1): (1) – 7(1) + 3 = -3, which is divisible by 3. Hence, P(1) is true. Let us assume that P(n) is true for some natural number n = k. P(k) = K³– 7k + 3 is divisible by 3 or, K³ – 7k + 3 = 3m, m∈ N ........(i) P(k+ 1 ):(k + 1)³ – 7(k + 1) + 3 = k³ + 1 + 3k(k + 1) – 7k— 7 + 3 = k -7k + 3 + 3k(k + 1)-6 = 3m + 3[k(k + 1) - 2] [Using (i)] = 3[m + (k(k + 1) – 2)], which is divisible by 3 Thus, P(k + 1) is true whenever P(k) is true. So, by the principle of mathematical induction P(n) is true for all natural numbers n.

Prove the statements by the Principle of Mathematical Induction :n3 – 7n + 3 is divisible by 3, for all natural numbers n.

Answer»

Let P(n): n³ – 7n + 3 is divisible by 3, for all natural numbers n.

Now P(1): (1) – 7(1) + 3 = -3, which is divisible by 3.

Hence, P(1) is true.

Let us assume that P(n) is true for some natural number n = k.

P(k) = K³– 7k + 3 is divisible by 3

or, K³ – 7k + 3 = 3m, m∈ N ........(i)

P(k+ 1 ):(k + 1)³ – 7(k + 1) + 3

= k³ + 1 + 3k(k + 1) – 7k— 7 + 3 = k -7k + 3 + 3k(k + 1)-6

= 3m + 3[k(k + 1) - 2] [Using (i)]

= 3[m + (k(k + 1) – 2)], which is divisible by 3 Thus, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for all natural numbers n.