Prove the following by using the principle of mathematical induction for all `n in N`:`1^3+2^3+3^3+dotdotdot+n^3=((n(n+1))/2)^2`

Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`1^3+2^3+3^3+dotdotdot+n^3=((n(n+1))/2)^2`
Answer» `P(n) : 1^(3) + 2^(3) + 3^(3) + "…. + n^(3) = (n^(2)(n+1)^(2))/(4)` For `n = 1`, L.H.S. `= 1^(3) = 1` and `R.H.S. = (1^(2).2^(2))/(4) = 1` Thus, `P(1)` is true. Let `P(n)` be true for some `n = k`. i.e, `1^(3) + 2^(3) + 3^(3) + "...." + k^(3) = (k^(2) (k+1)^(2))/(4) "....."(1)` Now, we have to prove that ` P(n)` is true for ` n = k + 1`. i.e, `1^(3) + 2^(3) + 3^(3) + ".... " +k^(3) + (k+1)^(3) = ((k+1)^(2)(k+2)^(2))/(4)` Adding `(k+1)^(3)` on both sides of `(1)`, we get `1^(3) + 2^(3) + 3^(3) + ".... " +k^(3) + (k+1)^(3)` `= (k^(2)(k+1)^(2))/(4)+(k+1)^(3)` `= ((k+1)^(2))/(4) (k^(2)+4(k+1))` `= ((k+1)^(2)(k+2)^(2))/(4)` Thus, `P (k+1)` is true whenever `P(k)` is true. Hence, by the principle of malthematical induction, statement `P(n)` is true for all natural numbers.