Introduction to Sequences
A sequence is an ordered list of numbers that follow a pattern. Each number in a sequence is called a term, denoted a₁, a₂, a₃, etc. Sequences are fundamental to understanding patterns, growth, and mathematical relationships.
Sequences appear everywhere: in financial calculations (compound interest), biological growth (cell division), physics (radioactive decay), and computer science (algorithm analysis).
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, denoted d.
General Form
An arithmetic sequence: a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...
where a₁ is the first term, d is the common difference, and n is the term number.
Example 1: Arithmetic Sequence
Find the 10th term of: 3, 7, 11, 15, ...
a₁ = 3, d = 4
a₁₀ = 3 + (10 - 1)(4) = 3 + 36 = 39
Example 2: Finding the Common Difference
Find d given: a₅ = 23, a₁ = 7
23 = 7 + 4d
4d = 16
d = 4
Arithmetic Means
Arithmetic means are the terms between two given terms of an arithmetic sequence.
Example 3: Finding Arithmetic Means
Insert 4 arithmetic means between 5 and 20.
We need: 5, __, __, __, __, 20
d = (20 - 5) / 5 = 3
Sequence: 5, 8, 11, 14, 17, 20
Arithmetic Series
A series is the sum of terms in a sequence. For an arithmetic series:
or equivalently:
Example 4: Sum of Arithmetic Series
Find the sum of the first 20 positive integers.
S₂₀ = 20(1 + 20) / 2 = 20(21) / 2 = 210
Example 5: Sum from 1 to 100
1 + 2 + 3 + ... + 100
S₁₀₀ = 100(1 + 100) / 2 = 5050
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms. This ratio is called the common ratio, denoted r.
General Form
A geometric sequence: a₁, a₁r, a₁r², a₁r³, ...
where a₁ is the first term, r is the common ratio, and n is the term number.
Example 6: Geometric Sequence
Find the 6th term of: 2, 6, 18, 54, ...
a₁ = 2, r = 3
a₆ = 2 · 3⁵ = 2 · 243 = 486
Example 7: Finding Common Ratio
Find r given: a₃ = 24, a₁ = 6
24 = 6 · r²
r² = 4
r = ±2
Geometric Means
Geometric means are the terms between two given terms of a geometric sequence.
Example 8: Finding Geometric Means
Insert 3 geometric means between 2 and 16.
We need 5 terms total: a₁ = 2, a₅ = 16
16 = 2 · r⁴
r⁴ = 8
r = 8^(1/4) = √2
Sequence: 2, 2√2, 4, 4√2, 16
Geometric Series
Finite Geometric Series
Example 9: Sum of Finite Geometric Series
Find the sum: 2 + 6 + 18 + 54 + 162
a₁ = 2, r = 3, n = 5
S₅ = 2(1 - 3⁵) / (1 - 3) = 2(1 - 243) / (-2) = 2(242) / 2 = 242
Infinite Geometric Series
An infinite geometric series converges only when |r| < 1:
Example 10: Infinite Geometric Series
Find the sum: 1 + 1/3 + 1/9 + 1/27 + ...
a₁ = 1, r = 1/3
|r| = 1/3 < 1, so the series converges
S∞ = 1 / (1 - 1/3) = 1 / (2/3) = 3/2
Example 11: Divergent Series
Find the sum: 1 + 3 + 9 + 27 + ...
a₁ = 1, r = 3
|r| = 3 > 1, so the series diverges (no finite sum)
Fibonacci Sequence
The Fibonacci sequence is defined recursively: each term is the sum of the two preceding terms.
Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Example 12: Fibonacci Terms
F₆ = F₅ + F₄ = 5 + 3 = 8
F₇ = F₆ + F₅ = 8 + 5 = 13
Fibonacci appears in nature: flower petals, pinecones, hurricanes, and galaxies follow Fibonacci patterns.
Special Sequences
Triangular Numbers
1, 3, 6, 10, 15, ... (arrangements of dots forming triangles)
Square Numbers
1, 4, 9, 16, 25, ... (n²)
Cubic Numbers
1, 8, 27, 64, 125, ... (n³)
Key Formulas Summary
| Type | Sequence (nth term) | Series (sum) |
|---|---|---|
| Arithmetic | aₙ = a₁ + (n-1)d | Sₙ = n(a₁ + aₙ)/2 |
| Geometric | aₙ = a₁ · rⁿ⁻¹ | Sₙ = a₁(1-rⁿ)/(1-r) |
| Geometric (∞) | — | S∞ = a₁/(1-r), |r|<1 |
Key Takeaways
- Arithmetic sequences add the common difference d each step
- Geometric sequences multiply by common ratio r each step
- Series sum terms; use appropriate formula for arithmetic or geometric
- Infinite geometric series converges only when |r| < 1
- Fibonacci uses recursion: each term = sum of previous two
Practice Sequences
Test your knowledge of sequences and series with our practice tests.
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