TO prove with mathematical induction follow these steps
1) Proof for n = 1
Just prove that it is true if we keep the value of n = 1
2)Assume for n= k
Assume that the given statement is correct for n=k
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in your question
Assume
1^3 +2^3 +.....k^3 = (k(k+1)/2)^2
3) Using step 2(the assumption we made) to Prove the statement for n = k+1
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in your case prove that
1^3 +2^3 + ....k^3 + (k+1)^3 = ((k+1)(k+2)/2)^2
using
1^3 +2^3 +.....k^3 = (k(k+1)/2)^2-------------(1)
So put (1) in the LHS thus
=(k(k+1)/2)^2 + (k+1)^3
Now simplify this LHS to get