# Math Help - Algebraic Proof

1. ## Algebraic Proof

So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

Question: Prove that the sum of two consecutive odd integers is always even.

My method;

Let two consecutive odd integers be $x$ and $x + 2$.

Hence, $x + (x + 2) = Even?$

$2x + 2 = Even$

Since twice anything is always even;

$Even + even = even$, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.

2. Originally Posted by SuperCalculus
So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

Question: Prove that the sum of two consecutive odd integers is always even.

My method;

Let two consecutive odd integers be $x$ and $x + 2$.

Hence, $x + (x + 2) = Even?$

$2x + 2 = Even$

Since twice anything is always even;

$Even + even = even$, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.
Prehaps you were expected to consider $2k+1~\&~2k+3$.
That makes clear that the integers are odd.

3. Originally Posted by Plato
Prehaps you were expected to consider $2k+1~\&~2k+3$.
That makes clear that the integers are odd.
Yeah, I'm aware that I proved it for all integers; but that's the point, ALL integers, and therefore including the odd ones, surely? =/.

4. Originally Posted by SuperCalculus
So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

Question: Prove that the sum of two consecutive odd integers is always even.

My method;

Let two consecutive odd integers be $x$ and $x + 2$.

Hence, $x + (x + 2) = Even?$

$Even + even = even$, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.
$2x + 2 = Even$

Since twice anything is always even;

$2x+2 = 2(x+1)$

since x+1(=m, suppose) is also an integer. Then 2m is an even number.

I would have started it this way:

Let n be any integet. Then $2n+1$ is always odd. The next consecutive integer will be $2n+1+2=2n+3$
so the sum is 4n+4= 4(n+1) which is even

5. Originally Posted by SuperCalculus
Yeah, I'm aware that I proved it for all integers; but that's the point, ALL integers, and therefore including the odd ones, surely? =/.
But it did said two consecutive odd integers.
Some graders can be very picky.