Arithmetic & Geometric Series

Introduction

A sequence is a list of terms that follow a pattern, like 3, 7, 11, 15. A series is what you get when you add those terms together: 3 + 7 + 11 + 15. Once a sequence has many terms, adding them one by one becomes slow and error-prone — series formulas let you find the total instantly, whether you're adding 10 terms or 10,000.

This lesson builds directly on arithmetic and geometric sequences, so it helps to be comfortable finding the common difference or common ratio before working through the formulas below.

Summing an Arithmetic Series

An arithmetic series is the sum of an arithmetic sequence — a sequence with a constant common difference, d.

Formula: Sₙ = n/2 (2a + (n - 1)d)

Where:

• Sₙ = sum of the first n terms
• a = first term
• d = common difference
• n = number of terms

There's an equivalent version that's handy when you already know the last term, l:

Formula: Sₙ = n/2 (a + l)

This works because pairing the first and last terms, the second and second-last, and so on, always gives the same pair-sum.

Worked Example: Find the sum of the first 10 terms of 3, 7, 11, 15, …

Step 1: Identify a = 3, d = 4, n = 10

Step 2: Substitute into Sₙ = n/2(2a + (n-1)d)

Step 3: S₁₀ = 10/2 (2×3 + 9×4) = 5 (6 + 36) = 5 × 42 = 210

Summing a Geometric Series

A geometric series is the sum of a geometric sequence — a sequence with a constant common ratio, r.

Formula (for r ≠ 1): Sₙ = a(1 - rⁿ) / (1 - r)

This can also be written as Sₙ = a(rⁿ - 1) / (r - 1), which avoids negative numbers when r is greater than 1.

Worked Example: Find the sum of the first 6 terms of 2, 6, 18, 54, …

Step 1: Identify a = 2, r = 3, n = 6

Step 2: Substitute into Sₙ = a(rⁿ - 1)/(r - 1)

Step 3: S₆ = 2(3⁶ - 1)/(3 - 1) = 2(729 - 1)/2 = 2(728)/2 = 728

Special case: if r = 1, every term is identical, so the formula above would divide by zero. Instead use Sₙ = na (simply n copies of the same term added together).

Sum to Infinity

Some geometric series can be summed even when they go on forever — but only if the terms shrink toward zero. This happens when the common ratio satisfies |r| < 1 (that is, r is between -1 and 1, not including either endpoint).

Formula: S∞ = a / (1 - r), valid only when |r| < 1

Worked Example: Find the sum to infinity of 8, 4, 2, 1, …

Step 1: Identify a = 8, r = 0.5. Since |0.5| < 1, the series converges.

Step 2: S∞ = 8 / (1 - 0.5) = 8 / 0.5 = 16

Intuitively, each new term adds a smaller and smaller amount, so the running total approaches — but never quite reaches — 16.

If |r| >= 1 (for example r = 2, or r = -1), the terms don't shrink, the series has no finite sum, and the sum-to-infinity formula cannot be applied.

Sigma Notation

Series are often written using sigma notation, where the Greek letter Σ means "sum of." For example, the sum of the first 5 terms of a sequence with nth term 3n + 2 can be written:

Σ (from n=1 to 5) of (3n + 2)

This means: substitute n = 1, 2, 3, 4, 5 into 3n + 2, and add the results together. Sigma notation is a compact way of describing exactly which terms to sum, without writing them all out.

Real-World Uses

Series show up whenever quantities accumulate over time or repeat with a pattern:

• Savings: Adding a fixed amount to an account each month is an arithmetic series; the total saved after n months uses the arithmetic sum formula.
• Compound growth: Repeated percentage increases (like compound interest) form a geometric sequence, and totalling contributions over several periods often involves a geometric series.
• Stacking and packing: Rows of objects that increase by a constant number (like tins stacked in a triangular display) form an arithmetic series.
• Bouncing ball problems: A ball that loses a fixed proportion of height on each bounce traces out a geometric series, and the sum-to-infinity gives the total distance travelled.

Common Mistakes

• Confusing a sequence (the list of terms) with a series (their sum)
• Using the arithmetic sum formula on a geometric sequence, or vice versa
• Forgetting the special case r = 1 in the geometric sum formula, which causes division by zero
• Applying the sum-to-infinity formula when |r| >= 1 — the series doesn't actually converge, so there is no valid sum
• Losing track of whether n counts the number of terms or the position of the last term

Tips and Tricks

• Always find a and d (or a and r) first, then substitute carefully — most errors come from substitution, not the formula itself
• Check an arithmetic sum by estimating: multiply the number of terms by the average of the first and last term
• For geometric series, decide early whether r is greater than 1 (growing) or between -1 and 1 (shrinking) — this tells you whether a sum to infinity even makes sense
• Sigma notation problems become easier if you write out the first few terms before summing