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College Algebra Unit 11: Sequences, series & binomial theorem

Sequences are ordered lists of numbers; series add them up. The binomial theorem expands powers of a binomial without multiplying it out.

Covers: Arithmetic/geometric sequences, sigma notation, factorials, binomial theorem

Study guide

A sequence is a function whose domain is the positive integers. Arithmetic sequences add a common difference: aₙ = a₁ + (n − 1)d. Geometric sequences multiply by a common ratio: aₙ = a₁rⁿ⁻¹. Sequences can also be defined recursively, giving each term from earlier ones.

Sigma notation writes sums compactly: Σᵢ₌₁ⁿ aᵢ means add aᵢ for i from 1 to n. An arithmetic series sums to Sₙ = n(a₁ + aₙ)/2. A finite geometric series sums to Sₙ = a₁(1 − rⁿ)/(1 − r). An infinite geometric series converges to a₁/(1 − r) only when |r| < 1; otherwise it has no finite sum.

Factorials count arrangements: n! = n(n − 1)···2·1, with 0! = 1 by definition. Combinations count selections: C(n, r) = n!/(r!(n − r)!), the number of ways to choose r items from n when order doesn't matter.

The binomial theorem expands powers of binomials: (x + y)ⁿ = Σ C(n, k)xⁿ⁻ᵏyᵏ, for k from 0 to n. The coefficients come from Pascal's triangle, where each entry is the sum of the two above it. To find one specific term, use C(n, k)xⁿ⁻ᵏyᵏ directly. For example, the coefficient of x²y in (x + y)³ is C(3, 1) = 3. Exponents in each term always add to n.

Sums can be split and scaled: Σ caᵢ = cΣ aᵢ, Σ (aᵢ + bᵢ) = Σ aᵢ + Σ bᵢ, and Σᵢ₌₁ⁿ c = nc. If a ball is dropped from height h and each bounce rises to a fraction r of the previous height, the total distance traveled is h + 2hr/(1 − r). Each rebound is traveled once up and once down.

For (ax − b)ⁿ, each term is C(n, k)(ax)ⁿ⁻ᵏ(−b)ᵏ, so the signs alternate and the coefficients include powers of a and b. The highest-degree term is (ax)ⁿ = aⁿxⁿ.

Key ideas

• Arithmetic sequences add the same difference d each time.
• Geometric sequences multiply by the same ratio r each time.
• Sigma notation Σ is shorthand for a sum.
• An infinite geometric series converges only when |r| < 1.
• n! = n·(n − 1)···1, and C(n, k) = n! / (k!(n − k)!) gives binomial coefficients.

Formulas and rules

Common mistakes

• Using the infinite-sum formula when |r| ≥ 1.
• Off-by-one errors: the nth term uses n − 1.

Words to know

nth term of an arithmetic sequence

aₙ = a₁ + (n − 1)d

nth term of a geometric sequence

aₙ = a₁·rⁿ⁻¹

Sum of an infinite geometric series

a₁/(1 − r), only if |r| < 1

Zero factorial

0! = 1 by definition.

Binomial coefficient C(n, r)

The number of ways to choose r of n items: n!/(r!(n − r)!).

Binomial theorem

(x + y)ⁿ = Σ C(n, k)xⁿ⁻ᵏyᵏ

Arithmetic series sum

Sₙ = n(a₁ + aₙ)/2

Finite geometric series sum

Sₙ = a₁(1 − rⁿ)/(1 − r)

Sequence

A function whose domain is the positive integers.

Recursive formula

Gives each term from earlier terms.

Sigma notation

Σ: the sum of terms for each index value.

Factorial

n! = n(n − 1)···2·1

Pascal's triangle

Each entry is the sum of the two above; rows give binomial coefficients.

Sample practice questions

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