According to the method of mathematical induction, one has to prove that statement
1) is correct for the initial value (in this task it is n=1)
2) assuming that the statement is valid for arbitrary n, prove its validity for (n+1).
Executing these steps, we obtain:
1) n = 1:
13=1,(21⋅2)2=1,1=1 statement is proved.
2) assuming that
13+23+...+n3=(2n(n+1))2 is correct, let us check it for (n+1) case:
13+23+...+n3+(n+1)3=?(2(n+1)(n+2))2 Simplifying the left-hand side of the expression, one can derive:
(13+23+...+n3)+(n+1)3=(2n(n+1))2+(n+1)3=
=(n+1)2(4n2+(n+1))=(n+1)24n2+4n+4=(2(n+1)(n+2))2, which coincides with the right-hand side of the assumption.
By the method of mathematical induction the statement is true for all natural values of n.
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