Question

What is wrong with this "proof"? "Theorem" For every positive integer n,n i =1 i = (n + 1 2 ) 2 /2. Basis Step: The formula is true for n = 1. Inductive Step: Suppose thatn i=1 i = (n + 1 2 ) 2 /2. Then n+1 i=1 i = ( n i=1 i) + (n + 1). By the induc-tive hypothesis, n+1 i=1 i = (n + 1 2 ) 2 /2 + n + 1 = (n 2 + n + 1 4 )/2 + n + 1 = (n 2 + 3n + 9 4 )/2 = (n + 3 2 ) 2 /2 =[ (n + 1) + 1 2 ] 2 /2, completing the induc-tive step.

Answer 1

Answer:

The basis step is not even true.

Step-by-step explanation:

The basis step would be for     n = 1.

If the basis step was true then you would have that.

$\sum\limits_{i=1}^{1} i = \frac{(1+1/2)^2}{2}$

Now. When you say

                                            $\sum\limits_{i=1}^{1} i$            

That's just a fancy way of writing    1.  

On the other hand    

$\frac{(1+1/2)^2}{2} = \frac{9}{4} = 2.25$

Therefore  

  $1 \neq 2.25$

And the basis hypothesis is false.