Using the principle of mathematical induction, prove that ` (7^(n)-3^(n))` is divisible by 4 for all ` n in N`.

Using the principle of mathematical induction, prove that ` (7^(n)-3^(

1.	Using the principle of mathematical induction, prove that ` (7^(n)-3^(n))` is divisible by 4 for all ` n in N`.
Answer» Let `P9n): (7^(n)-3^(n))` is divisible by 4. For n = 1, the given expression becomes `(7^(1)-3^(1))= 4`, which is divisible by 4. So, the given statement is true for n = 1 , i.e., P(1) is true. Let P(k) be true. Then, `P(k): (7^(k)-3^(k))` is divisible by 4. ` rArr (7^(k)-3^(k)) = 4m` for some natural number m. ....(i) Now, `{7^((k+1))-3^((k+1))}` `=7^((k+1))-73^(k)+73^(k)-3^((k+1))" "` [subtracting and adding `73^(k)`] ` = 7(7^(k)-3^(k))+3^(k)(7-3)` ` =(7xx4m)+43^(k)` [using (i)] ` = 4(7m+3^(k))`, which is clearly divisible by 4. ` :. P(k+1):{7^((k+1))-3^((k+1))}` is divisible by 4. Thus, P(k+1) is true, whenever P(k) is true. ` :. ` P(1) is true and P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction, it follows that `(7^(n)-3^(n))` is divisible by 4 for all values of ` n in N`.