Considering n = 1, we have that LHS = 1 = 1(1+1)/2 = RHS.
Now assume the proposition is true for n=k and consider
n=k+1.

LHS

=

1 + 2 + ... + k + (k + 1)

=

k(k + 1)/2 + (k + 1)

[induction hypothesis]

=

(k^{2} + 3k + 2)/2

[arithmetic]

=

(k + 1)(k + 2)/2

[factoring out]

=

(k + 1)((k + 1) + 1)/2

[arithmetic]

=

RHS

It would be very unwise to appeal to the above proof and claim that
the functions sum and sum' are indistinguishable. In
the first place, this is simply not true since they return different
answers for negative numbers. What we may claim based on the above
proof is that when both functions return a result for a positive
integer, it will be the same result. More information on using
induction to prove properties of functions may be found in
[MNV73].