Sequences and Series

A sequence is an ordered list; a series is the sum of its terms. Algebraic notation lets us describe a distant term or a large sum without writing every value.

Explicit and Recursive Rules

An explicit rule gives a_n directly from n. A recursive rule gives a starting term and explains how to obtain the next one.

Pattern	Explicit rule	Recursive rule
Arithmetic, common difference d	an = a1 + (n − 1)d	an = an−1 + d
Geometric, common ratio r	an = a1rn−1	an = r an−1

Worked example: identify and predict

For 7, 12, 17, 22, …, find the 30th term.

Common difference d = 5
a₁ = 7

aₙ = a₁ + (n − 1)d
a₃₀ = 7 + (30 − 1)(5)
     = 7 + 145
     = 152

Finite Arithmetic Series

Sₙ = n/2 [2a₁ + (n − 1)d]
or
Sₙ = n/2 (a₁ + aₙ)

Worked example: add an arithmetic sequence

Find 4 + 9 + 14 + … + 99.

99 = 4 + (n − 1)(5)
95 = 5(n − 1)
19 = n − 1
n = 20

S₂₀ = 20/2(4 + 99)
    = 10(103)
    = 1030

Finite and Infinite Geometric Series

Finite:   Sₙ = a₁(1 − rⁿ)/(1 − r), r ≠ 1
Infinite: S∞ = a₁/(1 − r), only when |r| < 1

An infinite geometric series converges only when repeated multiplication makes the terms approach zero. If |r| ≥ 1, no finite sum exists.

Worked example: repeating decimal

Write 0.272727… as a fraction using a series.

0.272727... = 0.27 + 0.0027 + 0.000027 + ...

a₁ = 27/100
r = 1/100

S∞ = a₁/(1 − r)
   = (27/100)/(99/100)
   = 27/99
   = 3/11

Sigma Notation

The symbol Σ means “sum.” The lower value gives the starting index and the upper value gives the final index.

Worked example: evaluate a sigma sum

Σ from k = 1 to 4 of (2k + 1)
= (2·1 + 1) + (2·2 + 1) + (2·3 + 1) + (2·4 + 1)
= 3 + 5 + 7 + 9
= 24

Practice Set

1. Find an explicit rule for 11, 8, 5, 2, …, then find a₂₅.
2. Find the sum of the first 40 terms of 3, 7, 11, ….
3. For 6, 3, 1.5, …, state r and find a₈.
4. Find the infinite sum 12 − 4 + 4/3 − ….
5. Evaluate Σ from k = 1 to 5 of k².

Answer Checks

1. d = −3, so a_n = 11 − 3(n − 1); a₂₅ = −61.
2. a₄₀ = 3 + 39(4) = 159; S₄₀ = 40(3 + 159)/2 = 3240.
3. r = 1/2; a₈ = 6(1/2)⁷ = 3/64.
4. a₁ = 12 and r = −1/3, so S_∞ = 12/[1 − (−1/3)] = 9.
5. 1 + 4 + 9 + 16 + 25 = 55.