Arithmetic Sequences and Finite Sums

Algebra Foundation

Algebra 2

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Algebra 2 · Unit 9 · Topic 9.2

Arithmetic Sequences and Finite Sums

For an arithmetic sequence with first term a₁ and difference d, aₙ = a₁ + (n − 1)d.

Definition and explanation

For an arithmetic sequence with first term a₁ and difference d, aₙ = a₁ + (n − 1)d. The sum of its first n terms is Sₙ = n(a₁ + aₙ)/2, where n is a positive integer.

Worked example

Sum the first 20 terms of 4, 7, 10, …. Here a₂₀ = 4 + 19(3) = 61, so S₂₀ = 20(4 + 61)/2 = 650. Pairing the first and last terms also gives ten pairs of 65.

Common mistake

The last term and the number of terms are different quantities in the sum formula.

Key takeaway

For an arithmetic sequence with first term a₁ and difference d, aₙ = a₁ + (n − 1)d.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.