Prove the following by the principle ofmathematical induction: `n^3-7n+3`is divisible 3 for all `n in N`.

Prove the following by the principle ofmathematical induction: `n^3-7n

1.	Prove the following by the principle ofmathematical induction: `n^3-7n+3`is divisible 3 for all `n in N`.
Answer» Let P(n) : `n^(3)-7n+3` is divisible by 3, for all natural number n. Step I We observe that P(1) is true. `P(1)=(1)^(3)-7(1)+3` =11-7+3 -3, which is divisible by 3. Hence, P(1) is true. Step II Now, assume that P(n) is true for n=k. `P(k+1):(k+1)^(3)-7(k+1)+3` `=k^(3)+1+3k(k+1)-7k-7+3` `=k^(3)-7k+3+3k(k+1)-6` `=3q+3[k(k+1)-2]` Hence, P(k+1) is true whenever P(k) is true. [from stem II] So, by the principle of mathematical induction P(n): is true for all natural number n.