Sequences & Series

Series & Summation Notation

Finding the sum of a finite arithmetic or geometric series using summation notation.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Geometric Sequences

Adding a Sequence Without Adding One Term at a Time

Before reading on, think about finding 1 + 2 + 3 + ... + 100 without actually adding 100 numbers one at a time. A famous trick, often credited to a young Carl Friedrich Gauss, involves pairing numbers from opposite ends of the list — try pairing the first and last, the second and second-to-last, and see what pattern emerges.

Definition — Series and Summation Notation

A series is the sum of a sequence's terms. Summation (sigma) notation, Σ, compactly represents a series: the expression below and above the Σ give the starting and ending index, and the formula to the right gives each term.

Pairing 1 with 100, 2 with 99, 3 with 98, and so on, each pair sums to exactly 101 — and there are 50 such pairs (half of 100 terms). The total is 50 × 101 = 5,050. Generalizing this pairing trick to any arithmetic series of n terms, first term a₁, last term aₙ: there are n/2 pairs, each summing to (a₁ + aₙ), giving the arithmetic series sum formula.

S_n = \dfrac{n}{2}(a_1 + a_n)

Worked Example — Finding an Arithmetic Series Sum

Find the sum of the first 20 terms of the arithmetic sequence 4, 7, 10, .... First find a₂₀: a₂₀ = 4 + (20 − 1)(3) = 4 + 57 = 61. Apply the sum formula: S₂₀ = (20/2)(4 + 61) = 10 × 65 = 650.

A geometric series sum uses a different trick: call the sum S, multiply the entire sum by r to get rS, and subtract. Every term except the very first and very last cancels between S and rS, since each shifts by one power of r — leaving a simple equation that solves directly for S.

S_n = a_1 \cdot \dfrac{1 - r^{n}}{1 - r} \quad (r \neq 1)

Worked Example — Finding a Geometric Series Sum

Find the sum of the first 8 terms of the geometric sequence 3, 6, 12, .... Here a₁ = 3, r = 2, n = 8: S₈ = 3 · (1 − 2⁸)/(1 − 2) = 3 · (1 − 256)/(−1) = 3 · 255 = 765.

Worked Example — Evaluating Summation Notation

Evaluate Σ (from k=1 to 5) of (2k + 1). Expand each term: k=1 gives 3, k=2 gives 5, k=3 gives 7, k=4 gives 9, k=5 gives 11. Sum: 3 + 5 + 7 + 9 + 11 = 35.

Tip

For a small number of terms in summation notation, expanding every term individually (as in the last example) is often faster and less error-prone than trying to force-fit an arithmetic or geometric series formula.

Common Mistakes

Using the arithmetic series sum formula on a geometric series, or vice versa, without checking which type of sequence is actually involved.
Confirm whether consecutive terms share a constant difference (arithmetic) or a constant ratio (geometric) before choosing which sum formula applies — they are not interchangeable.
Miscounting the number of terms in a series, especially off by one when the summation's bounds are inclusive on both ends.
Summation notation from k = 1 to k = 5 includes both endpoints, giving exactly 5 terms — carefully count inclusive bounds rather than assuming the upper bound alone gives the term count.

Key Takeaways

The arithmetic series sum formula comes from pairing terms from opposite ends, each pair summing to the same total.
The geometric series sum formula comes from subtracting a shifted copy of the sum from itself, canceling every middle term.
Summation notation compactly represents a series, and can always be expanded term by term when that's simpler.

Summary

Series sum formulas, derived rather than memorized, close out sequences and series. The final unit extends trigonometric ratios beyond the right triangle, to any angle at all.

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