How to Prove an Arithmetic Sequence Identity

Through my online math tutoring, I work with students in Winnipeg and across Canada. A student recently brought me this question about arithmetic sequences. The key is to use the common difference to rewrite the terms.

The Student’s Question

If a, b, c, and d are four consecutive terms of an arithmetic sequence, prove that:

2(b − c)² = bc − ad

Start with the Common Difference

In an arithmetic sequence, we add the same amount to move from one term to the next. Call this common difference r, since d already represents the fourth term.

The four terms can therefore be written as:

a, a + r, a + 2r, a + 3r

To prove the identity, we’ll substitute these expressions and show that both sides simplify to the same result.

Simplify Each Side

Left-Hand Side: 2(b − c)²

Since b and c are consecutive terms, b − c = −r. Therefore:

2(b − c)²= 2(−r)²= 2r²

Right-Hand Side: bc − ad

Substitute, expand, and combine like terms:

bc − ad= (a + r)(a + 2r) − a(a + 3r)= a² + 3ar + 2r² − a² − 3ar= 2r²

Both sides equal 2r², so we have proved that:

2(b − c)² = bc − ad

Why This Proof Works

We haven’t chosen specific values for a or r, so the algebra applies to any four consecutive terms of an arithmetic sequence.

When approaching a proof like this, start with what the sequence tells you. Writing each term using the common difference makes the relationship much easier to see.

Math Tutoring for Winnipeg Students

If arithmetic sequences or algebraic proofs feel confusing, individual tutoring can help identify the step that needs more attention.

At Tutor Advance, I offer online math tutoring for Winnipeg students and students across Canada, with clear explanations and practice tailored to each student’s needs. As your math tutor, I can help you understand why each step works and apply that reasoning independently.

Contact Tutor Advance to discuss your math course, learning goals, and current tutoring availability.