13. Show that 9" cannot end with 2 for any natural number.

1.	13. Show that 9" cannot end with 2 for any natural number.
Answer» Let, P(n) denotes the event that 9^n can not end with the digit 2 When, n=1, P(1): 9^1 = 9 does not end with 2. Let us assume that the statement is true for n=k;Then, P(k): 9^k does not end with 2. Now for n=k+1, P(k+1): = 9^k × 9^1 which does not end with 2 since 9^k and 9 both does not end with 2. Therefore, assuming P(n) is true for n=k we have that P(n) is true for n=k+1. Thus, by the principle of mathematical induction, P(n): 9^n does not end with 2.

Answer»

Let, P(n) denotes the event that 9^n can not end with the digit 2

When, n=1, P(1): 9^1 = 9 does not end with 2.

Let us assume that the statement is true for n=k;Then, P(k): 9^k does not end with 2.

Now for n=k+1, P(k+1): = 9^k × 9^1 which does not end with 2 since 9^k and 9 both does not end with 2.

Therefore, assuming P(n) is true for n=k we have that P(n) is true for n=k+1.

Thus, by the principle of mathematical induction, P(n): 9^n does not end with 2.

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