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

Options

A. 3
B. 4
C. 5
D. 6

Answer

B

Explanation

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))-7*3^(k)+7*3^(k)-3^((k+1))" "` [subtracting and adding `7*3^(k)`]
` = 7(7^(k)-3^(k))+3^(k)(7-3)`
` =(7xx4m)+4*3^(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`.