Algebraic Proof Practice Questions

Write down an expression, in terms of $n$, for an even number.

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

Prove that the shaded area, in cm$^2$, is always a multiple of $24$.

Prove that every term of the sequence with $n$th term $A_n - B_n$ is a multiple of $4$.

Write $x^2 - 6x + 11$ in the form $(x + a)^2 + b$, where $a$ and $b$ are integers.

Hence prove that $x^2 + 11 > 6x$ for all real values of $x$.

Show that Rhys is wrong by giving a counter-example.

Prove that $n^2 + n + 1$ is an odd number for every positive integer $n$.

Factorise $f(n)$ fully.

Hence prove that $f(n)$ is a multiple of $6$ for every positive integer $n$.