Skill: series

Series & Summation Notation

From sigma notation to series formulas: expand and evaluate sums, and add up arithmetic and geometric series with confidence.

1 Reading sigma notation

The Greek letter Σ (sigma) is shorthand for “add them all up.” Every sigma has four parts.

Anatomy of a sum

∑_k=mⁿ a_k = a_m + a_m+1 + … + a_n

k is the index (the counter), m the lower limit, n the upper limit, and a_k the summand (the rule for each term). The number of terms is n − m + 1 — the index is not the term count.

Expand ∑_k=2⁵ (3k − 1)

k runs 2, 3, 4, 5 — that is 5 − 2 + 1 = 4 terms: (3·2−1) + (3·3−1) + (3·4−1) + (3·5−1) = 5 + 8 + 11 + 14 = 38.

2 Summing an arithmetic series

Pair the first term with the last, the second with the second-to-last — every pair sums to the same total.

Arithmetic series sum

S_n = n2(a₁ + a_n) = n2(2a₁ + (n − 1)d)

Use the left form when you know the last term a_n; use the right form when you know d. Both say the same thing: n copies of the average of the first and last terms.

Sum 4 + 7 + 10 + … + 61 (20 terms)

a₁ = 4, d = 3, n = 20. S₂₀ = 20/2 · (2·4 + 19·3) = 10 · (8 + 57) = 10 · 65 = 650.

3 Summing a geometric series

Multiply the whole sum by r, subtract, and almost everything cancels — leaving a clean formula.

Geometric series sum (r ≠ 1)

S_n = a₁ · 1 − rⁿ1 − r

The exponent is n here (not n − 1): the first term uses r⁰ = 1, so the sum runs through rⁿ⁻¹, and the formula carries one more power. If r = 1, every term equals a₁ and S_n = n · a₁.

Sum 2 + 6 + 18 + 54

a₁ = 2, r = 3, n = 4. S₄ = 2 · (1 − 3⁴)/(1 − 3) = 2 · (1 − 81)/(−2) = 2 · 40 = 80.

4 Worked examples

Count, expand, and sum — each claim verified independently.

Example 1 Count the terms

How many terms does ∑_k=3¹⁰ a_k have?

Terms: 10 − 3 + 1 = 8. The index starts at 3, so there are not 10 terms.

Example 2 Evaluate a sigma

Evaluate ∑_k=1⁵ (2k + 3).

Terms: 5, 7, 9, 11, 13. Sum = 45.

Example 3 Arithmetic sum from d

Find the sum of the first 20 terms with a₁ = 4, d = 3.

S₂₀ = 10 · (8 + 57) = 650.

Example 4 Arithmetic sum from a_n

An arithmetic series has a₁ = 5, a₁₀ = 50. Find S₁₀.

S₁₀ = 10/2 · (5 + 50) = 5 · 55 = 275.

Example 5 Geometric sum

Find the sum: a₁ = 2, r = 3, n = 4.

S₄ = 2(1 − 81)/(1 − 3) = 80.

Example 6 Geometric sum

Find the sum: a₁ = 5, r = 2, n = 6.

S₆ = 5(1 − 64)/(1 − 2) = 5 · 63 = 315.

Example 7 A single term of a sigma

In ∑_k=1⁸ k², what is the value of the 4th term?

Substitute k = 4: 4² = 16.

5 Common mistakes

Two errors that break series problems.

Confusing the index with the number of terms

Wrong: ∑_k=3¹⁰ has 10 terms.
Right: count them: upper − lower + 1 = 10 − 3 + 1 = 8 terms.

Using n instead of n − 1 in the exponent for geometric series

Wrong: the first term of a₁rⁿ thinking.
Right: the first term uses exponent 0: a₁ · r⁰ = a₁. The nth term is a₁rⁿ⁻¹; the sum formula carries rⁿ.

6 Key vocabulary

Words to know

Sigma (Σ) notation — compact notation for a sum: index, limits, and summand.
Index — the counter variable (usually k); not the number of terms.
Partial sum (S_n) — the sum of the first n terms.
Arithmetic series — sum of an arithmetic sequence: S_n = n/2 · (a₁ + a_n).
Geometric series — sum of a geometric sequence: S_n = a₁(1 − rⁿ)/(1 − r).

7 Quick check

Try these before moving on — click to reveal each answer.

How many terms are in ∑_k=4¹⁵ a_k?

15 − 4 + 1 = 12 terms.

Evaluate ∑_k=1⁴ (3k − 2).

Terms: 1, 4, 7, 10. Sum = 22.

Find S₅ for a₁ = 3, r = 2 (geometric).

S₅ = 3(1 − 32)/(1 − 2) = 3 · 31 = 93.

8 Next steps

Now drill the skill with endless randomized problems.

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