Algebra · step 12 of 13

Arithmetic & geometric sequences and series

An arithmetic sequence adds a constant difference d each step, giving aₙ = a + (n−1)d and Sₙ = n/2 (2a + (n−1)d). A geometric sequence multiplies by a constant ratio r, giving aₙ = ar^(n−1) and Sₙ = a(1 − rⁿ)/(1 − r). The infinite geometric sum a/(1 − r) exists exactly when |r| < 1 — a condition that must be checked, not assumed.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identify the typeConstant DIFFERENCE means arithmetic; constant RATIO means geometric. Compute a few differences and ratios to decide.
Extract a and d or rFrom two given terms, set up simultaneous equations in a and the parameter.
Apply the right sum formulaArithmetic sums use the average of the first and last term times n; geometric sums use the ratio formula.
Check convergence before summing to infinityS∞ = a/(1 − r) requires |r| < 1. Otherwise the series diverges and there is no sum.

Worked example

A geometric series has third term 12 and sixth term 96. Find r, a, and the sum of the first 8 terms.

ar² = 12 and ar⁵ = 96. Divide: r³ = 8, so r = 2.
From ar² = 12: 4a = 12, so a = 3.
S₈ = a(rⁿ − 1)/(r − 1) = 3(2⁸ − 1)/(2 − 1).
= 3(256 − 1) = 3 · 255.

Answer. r = 2, a = 3, and S₈ = 765.

Where marks get dropped

These are the specific errors that cost credit on arithmetic & geometric sequences and series questions — QED's rubric penalises each of them separately.

Using aₙ = a + nd instead of a + (n−1)d. The first term corresponds to n = 1, so the difference is applied n − 1 times.
Applying the infinite sum formula when |r| ≥ 1. With r = 2 the series diverges and a/(1−r) returns a meaningless negative number.
Confusing the sequence with the series. The nth term and the sum of n terms are different questions.

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Arithmetic & geometric sequences and series — frequently asked questions

Why does |r| < 1 give a finite sum?

Because rⁿ → 0, so Sₙ = a(1 − rⁿ)/(1 − r) → a/(1 − r). When |r| ≥ 1 the terms do not shrink and the partial sums grow without bound.

How do I turn a recurring decimal into a fraction?

Treat it as a geometric series. 0.333… = 3/10 + 3/100 + … with a = 3/10 and r = 1/10, giving (3/10)/(9/10) = 1/3.

What is the sum of an arithmetic series in words?

Number of terms times the average of the first and last term — which is why Gauss could add 1 to 100 instantly as 100 × 50.5.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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