1³ + 2³ + 3³ + …. +n³ = .

Let P(n) = 1³ + 2³ + 3³ + …. +n³ =
P(1) is true
Assume P(k) is true.
1³ + 2³ + 3³ +…..+ K³ =
To Prove P(K+1) is true using P(k)
P(K + 1) = 1³ + 2³ + 3³ + K³ + (K + 1)³ = ……….(1)
L.H.S of (1)
1³ + 2³ + 3³ + K³ + (K + 1)³ = + (K +1)³
                                                 = + (K +1)³
                                                 =
                                                 =
                                                 =
                                                 =
                                         =
R.H.S of ( 1).
Therefore P(K + 1) is proved
Hence by the principle of 4 mathematical induction P(n) is true for all the Values of n,
n N
Hence proved.