Let the given statement be P(n) i.e.,
P (n) : 13+23+33 + ... + n3 = (

n(n+1)

2

)2
First we prove that the statement is true for n = 1.
P (1) : = (

1(1+1)

2

)2 = (

1×2

2

)2 = (1)2 = 1 , which is true
Assume that P(k) is true for some positive integer k, i.e.,
13+23+33 + ... + k3 = (

k(k+1)

2

)2 ... (i)
We shall now prove that P(k + 1) is also true. For this we have to prove
13+23 + ... + k3+(k+1)3 = (

(k+1)[(k+1)+1]

22

)
L.H.S = 13+23 + ... + k3+(k+1)3 = (

k(k+1)

2

)2 + (k+1)3 (By using (i))
= (k+1)2(

k2

4

+k+1) = (k+1)2(

k2+4k+4

4

)
= (k+1)2(

(k+2)2

2

) =

(k+1)2((k+1)+1)2

(2)2

= (

(k+1)[(k+1)+1]

22

)
∴ P(k + 1) is true, whenever P(k) is true.
Hence, by the principle of mathematical induction, P(n) is true ∀ n ∈ N.