There is something wrong with this proof.

Prove:

1+2+ … + n = n(n+1)/2

Base case: n =1

1 = 1(1+1)/2 = 1

Inductive case:

Prove: 1+2+ … + n + (n+1) = (n+1)(n+2)/2

n(n+1)/2 + (n+1) = (n+1)(n+2)/2

n(n+1) + 2(n+1) = (n+1)(n+2)

(n+2)(n+1) = (n+1)(n+2)

The error is that the inductive case begins with the inductive property - the thing we want to prove - and from it derives an obviously true conclusion. The convention in a proof is that each step follows from the preceding steps. Therefore, it should begin with what is assumed to be the case, and end with what you are trying to prove. For example:

Inductive case:

Prove: 1+2+ … + n + (n+1) = (n+1)(n+2)/2

Inductive hypothesis: 1+2+ … + n = n(n+1)/2

1+2+ … + n + (n+1) = n(n+1)/2 + (n+1) [from inductive hypothesis and addition of equals]

1+2+ … + n + (n+1) = n(n+1)/2 + 2(n+1)/2 [multiply term by 2/2]

1+2+ … + n + (n+1) = [n(n+1) + 2(n+1)]/2 [distributive law]

1+2+ … + n + (n+1) = (n+1)(n+2)/2 [factoring]

QED

To see the problem more concretely, consider the following:

Prove (incorrectly) that P holds.

Hypothesis: P

P & true [conjoining with true does not affect the value of a predicate]

P is true [if a conjunction is true, each conjunct is true]

There is another way to fix the inductive proof, but it makes the proof harder to understand. This is to explicitly state that each step follows from the next step, rather than the previous step. We still need to distinguish the inductive hypothesis from the inductive property being proven.

Inductive case:

Prove: 1+2+ … + n + (n+1) = (n+1)(n+2)/2

Inductive hypothesis: 1+2+ … + n = n(n+1)/2

Proof:

1+2+ … + n + (n+1) = (n+1)(n+2)/2

is true by substitution and the inductive hypothesis if

n(n+1)/2 + (n+1) = (n+1)(n+2)/2

which is true by multiplication by equals if

n(n+1) + 2(n+1) = (n+1)(n+2)

which is true by the multiplication if

(n+2)(n+1) = (n+1)(n+2)

which is true by the communicative law.

This proof is clearly harder to understand. Finally, it is sometimes possible to reduce an entire proof to a single step, so the forward and backward directions are the same. You can get away with this if your proof can be represented as a series of equal expressions:

Inductive case:

Prove: 1+2+ … + n + (n+1) = (n+1)(n+2)/2

Inductive hypothesis: 1+2+ … + n = n(n+1)/2

Proof:

1+2+ … + n + (n+1) =

n(n+1)/2 + (n+1) =

n(n+1)/2 + 2(n+1)/2 =

[n(n+1) + 2(n+1)]/2 =

(n+1)(n+2)/2

Note the equalities at the end of each line: they mean this is whole proof is a single compound statement, not a sequence of steps. Each line contains a numeric expression, not a separate assertion.