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Algebra 2 Unit 7: Sequences & series

Sequences are ordered lists; series are their sums. Arithmetic ones add a constant, geometric ones multiply by one.

Covers: Arithmetic and geometric sequences, recursive and explicit formulas, sigma notation, finite and infinite sums

Study guide

A sequence is an ordered list of numbers. An arithmetic sequence adds a common difference d each time: aₙ = a₁ + (n − 1)d. A geometric sequence multiplies by a common ratio r each time: aₙ = a₁·rⁿ⁻¹. To identify which you have, check for constant differences or constant ratios.

Sequences can be defined two ways. A recursive formula gives the first term and a rule for getting each term from the one before, like a₁ = 5, aₙ = aₙ₋₁ + 3. An explicit formula gives any term directly from n, like aₙ = 5 + 3(n − 1). Explicit formulas are faster for finding far-away terms.

A series is the sum of a sequence's terms. Sigma notation, Σ, compactly writes a sum: the index starts at the bottom value and runs to the top value, and you add the expression for each index value.

An arithmetic series with n terms sums to Sₙ = n(a₁ + aₙ)/2, the number of terms times the average of the first and last (so 1 + 2 + … + 100 = 5050). A finite geometric series sums to Sₙ = a₁(1 − rⁿ)/(1 − r). An infinite geometric series converges only when |r| < 1, and then its sum is a₁/(1 − r). If |r| ≥ 1 the terms don't shrink and the series has no finite sum.

Geometric series model repeated payments or deposits and repeated bounces. Identify a₁, the common ratio r and the number of terms n before using Sₙ = a₁(1 − rⁿ)/(1 − r). A geometric sequence can be written recursively as a₁ = first term and aₙ = r·aₙ₋₁. Its explicit form is aₙ = a₁·rⁿ⁻¹.

Key ideas

• Arithmetic: aₙ = a₁ + (n − 1)d.
• Geometric: aₙ = a₁·rⁿ⁻¹.
• Recursive formulas define each term from the previous one.
• Sigma notation Σ adds terms over an index.
• Infinite geometric series converge only when |r| < 1.

Formulas and rules

Common mistakes

• Using n instead of n − 1 in the nth-term formula.
• Applying the infinite sum formula when |r| ≥ 1.

Words to know

Arithmetic nth term

aₙ = a₁ + (n − 1)d

Geometric nth term

aₙ = a₁·rⁿ⁻¹

Arithmetic series sum

Sₙ = n(a₁ + aₙ)/2

Infinite geometric sum

a₁/(1 − r) when |r| < 1

Recursive vs explicit formula

Recursive uses the previous term; explicit uses n directly.

Sequence

An ordered list of numbers.

Common difference

The constant added each step in an arithmetic sequence.

Common ratio

The constant multiplier in a geometric sequence.

Series

The sum of a sequence's terms.

Sigma notation

Σ: add the expression for each index value from bottom to top.

Finite geometric sum

Sₙ = a₁(1 − rⁿ)/(1 − r)

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.