15.5 Algebraic proof — Learn

Prerequisites: expand brackets, collect like terms, and use algebraic notation for integers.

Objectives: represent numbers with variables, build a chain of valid equalities, and prove divisibility or an identity. This section teaches algebraic proof.

A proof uses general reasoning to show that a statement is true for every value in its stated set. Consecutive integers differ by one. An even integer can be written as $2n$, where $n$ is an integer.

Example 1. Prove that the sum of three consecutive integers is divisible by $3$.

Let the integers be $n$, $n+1$, and $n+2$. Add them: $n+(n+1)+(n+2)$. Collect terms: $3n+3$. Factorise: $3(n+1)$. Since $n+1$ is an integer, the sum is a multiple of $3$.

Example 2. Prove that the difference between the squares of consecutive integers is odd.

Let the integers be $n$ and $n+1$. Subtract the squares: $(n+1)^2-n^2$. Expand: $n^2+2n+1-n^2$. Simplify: $2n+1$. This is twice an integer plus one, so it is odd.

Watch out. Testing examples can suggest a result, but it does not prove a statement for all integers. Use a general integer and explain why the final form has the required property.

Progress check

Write the first three even integers using $n$. What expression represents their sum?

You should now be able to

• explain the main method in this section;
• show each algebraic step and check your result;
• state any excluded values or conditions.