Algebraic Proof Practice Questions

$2n$ is an even number. Write down an expression, in terms of $n$, for the next even number after $2n$.

Prove algebraically that the sum of any three consecutive even numbers is always a multiple of $6$.

Find the value of $a$, the value of $b$ and the value of $c$.

Hence prove that $3x^2 - 18x + 32$ is positive for all values of $x$.

Using algebra, prove that $0.\dot{7}\dot{2} \div 0.\dot{4}$ is equal in value to $\frac{18}{11}$.

The diagram shows a rectangle. $n$ is an integer.

Prove that the area of the rectangle is always $1$ less than a square number.

Explain what is wrong with Marek's proof.

Write a correct proof that the sum of any two different odd numbers is always even.

Prove algebraically that the straight line with equation $y = 2x + 5$ does not intersect the circle with equation $x^2 + y^2 = 4$.

$n$ is a positive integer.

Prove algebraically that the difference between the cubes of any two consecutive integers is always $1$ more than a multiple of $6$.