Algebraic proof questions and answers

Worked solution

1. $2n = 2 \times n$ is a multiple of 2 for any integer $n$.

Worked solution

1. $2n$ is even, so $2n + 1$ is odd.

Worked solution

1. $2n$ is even, so $2n - 1$ is always odd.

Worked solution

1. Each integer is one more than the one before: $n$, $n + 1$, $n + 2$.

Worked solution

1. $3(x + 4) = 3x + 12$
2. $-2(x - 1) = -2x + 2$
3. $3x + 12 - 2x + 2 = x + 14$ ✓

Worked solution

1. $n + (n + 1) + (n + 2) = 3n + 3$
2. $= 3(n + 1)$
3. $3 \times$ an integer, so it's a multiple of 3.

Worked solution

1. $(2n + 1) + (2n + 3) = 4n + 4$
2. $= 4(n + 1)$, a multiple of 4.

Worked solution

1. $(n + 3)^2 = n^2 + 6n + 9$
2. $(n - 3)^2 = n^2 - 6n + 9$
3. Subtract: $12n$ ✓

Worked solution

1. $5(x + 2) = 5x + 10$
2. $-3(x - 4) = -3x + 12$
3. $5x + 10 - 3x + 12 = 2x + 22$ ✓

Worked solution

1. Let the even numbers be $2m$ and $2n$, where $m$ and $n$ are integers.
2. $2m + 2n = 2(m + n)$
3. $m + n$ is an integer, so the sum is a multiple of 2: it's even.

Worked solution

1. $(2m + 1)(2n + 1) = 4mn + 2m + 2n + 1$
2. $= 2(2mn + m + n) + 1$
3. That's one more than a multiple of 2, so it's odd.

Worked solution

1. $(2n + 1)^2 = 4n^2 + 4n + 1$
2. $(2n - 1)^2 = 4n^2 - 4n + 1$
3. Difference $= 8n = 8 \times n$, a multiple of 8.

Worked solution

1. $n^2 - n = n(n - 1)$
2. $n - 1$ and $n$ are consecutive, so one of them is even.
3. Even × anything is even.

Worked solution

1. $(2n + 3)^2 = 4n^2 + 12n + 9$
2. $(2n + 1)^2 = 4n^2 + 4n + 1$
3. Difference $= 8n + 8 = 8(n + 1)$, a multiple of 8.

Worked solution

1. $n^2 + (n + 1)^2 = n^2 + n^2 + 2n + 1$
2. $= 2n^2 + 2n + 1 = 2(n^2 + n) + 1$
3. That's one more than a multiple of 2, so it's always odd.