Use mathematical induction to prove the statement.
13 + 23 + 33 + .... + n3 = ,
∀ n ∈ N.
Sol : Let P(n) be the statement :
13 + 23 + 33 + .... + n3 =
Since the formula is true for n = 1.
Assume the statement P(n) is true for n = k.
i.e., 13 + 23 + 33 + .... + k3 =
We show that the formula is true for n = k + 1,
i.e., we show that S(k + 1) =
(where S(k) = 13 + 23 + 33 + .... + k3)
we observe that S(k + 1) = 13 + 23 + 33 + .... + k3 + (k + 1)3 = S(k) + (k + 1)3.
Since, S(k) = ,
we have
∴ The formula holds for n = k + 1.
∴ By the principle of mathematical induction, P(n) is true for all n ∈ N.
i.e., the formula 13 + 23 + 33 + .... + n3 =
is true for all n ∈ N.