Answer
The Basis Step is wrong
Work Step by Step
The question asks us 'What is wrong with this “proof”?'
What's wrong is the base case. It asserts that the formula is true for n = 1.
However,
LHS = $Σ^{1}_{i=1}i $ = 1
whereas
RHS = $\frac{(n+\frac{1}{2})^2}{2}|_{n=1}$ = $\frac{(1+\frac{1}{2})^2}{2}$
or RHS = $\frac{9}{8}$
Thus LHS$\ne$ RHS and hence the base case fails.
Note that $Σ^{n}_{i=1}i =\frac{n(n+1)}{2}$
