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Step 13: Algebra 2

Sequences and series

Step 13 · Algebra 2 topics

Write rules for arithmetic and geometric sequences, find any term, and add up finite and infinite series.

Arithmetic sequences

An arithmetic sequence adds the same number, the common difference d, each time.

an=a1+(n−1)da_n = a_1 + (n - 1)d

Example: The nth term

Find the 20th term of 7, 11, 15, 19, …

  1. a₁ = 7 and d = 4.
  2. a₂₀ = 7 + 19 × 4 = 83.

Answer83

Example: Finding the position

Which term of 5, 8, 11, … is 92?

  1. 5 + (n − 1) × 3 = 92.
  2. (n − 1) × 3 = 87, so n − 1 = 29.

AnswerThe 30th term

7, 11, 15, 19, 23, 27: an arithmetic sequence grows by the same step each time.

Geometric sequences

A geometric sequence multiplies by the same number, the common ratio r, each time.

an=a1⋅r n−1a_n = a_1 \cdot r^{\,n - 1}

Example: The nth term

Find the 8th term of 3, 6, 12, 24, …

  1. a₁ = 3 and r = 2.
  2. a₈ = 3 × 2⁷, which is 3 × 128 = 384.

Answer384

Example: A shrinking sequence

Find the next two terms of 800, 400, 200, … and its 10th term.

  1. r = 0.5, so the next terms are 100 and 50.
  2. a₁₀ = 800 × 0.5⁹, which is 800 ÷ 512 = 1.5625.

Answer100 and 50; the 10th term is 1.5625

Sums of series

A series is the sum of the terms of a sequence. Sigma notation, Σ, writes it compactly.

Sn=n(a1+an)2Sn=a1(1−rn)1−rS_n = \frac{n(a_1 + a_n)}{2} \qquad S_n = \frac{a_1(1 - r^{n})}{1 - r}

Example: An arithmetic series

Find 1 + 2 + 3 + … + 100.

  1. Pair the ends: 1 + 100, 2 + 99, and so on, each making 101.
  2. There are 50 pairs: 50 × 101 = 5,050.

Answer5,050

Example: A geometric series

Find 2 + 6 + 18 + … to 6 terms.

  1. S6=2(1−36)1−3=2(1−729)−2S_6 = \frac{2(1 - 3^{6})}{1 - 3} = \frac{2(1 - 729)}{-2}
  2. That is 728.

Answer728

Example: An infinite series

Find 8+4+2+1+⋯8 + 4 + 2 + 1 + \cdots.

  1. When r is between −1 and 1, the sum settles down: S=a11−r=80.5S = \frac{a_1}{1 - r} = \frac{8}{0.5}.

Answer16

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Find the 15th term of 2, 9, 16, 23, …

    Show answer

    100

  2. Find the common ratio of 81, 27, 9, …

    Show answer

    13\frac{1}{3}

  3. Find the 6th term of 5, 10, 20, …

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    160

  4. Find 3 + 7 + 11 + … + 43.

    Show answer

    253

  5. Find 1 + 3 + 9 + 27 + 81.

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    121

  6. Find 12 + 6 + 3 + … going on forever.

    Show answer

    24

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