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:
statement is proved.
2) assuming that
is correct, let us check it for (n+1) case:
Simplifying the left-hand side of the expression, one can derive:
"=(n+1)^2 \\left(\\frac{n^2}{4} + (n+1) \\right) = (n+1)^2 \\frac{n^2 +4n+4}{4} = \\left(\\frac{(n+1)(n+2)}{2}\\right)^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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