Lesson 1 • 45 min read
Sequences
In This Lesson
1Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference (d).
nth Term Formula
aₙ = a₁ + (n - 1)d
aₙ
nth term
a₁
first term
d
common difference
Example 1: Find the 10th term of 3, 7, 11, 15, ...
a₁ = 3, d = 7 - 3 = 4
a₁₀ = 3 + (10 - 1)(4)
a₁₀ = 3 + 36
a₁₀ = 39
Example 2: Find d if a₁ = 5 and a₈ = 33
33 = 5 + (8 - 1)d
33 = 5 + 7d
28 = 7d
d = 4
Finding Common Difference
d = a₂ - a₁ = a₃ - a₂ = ... (any term minus the previous)
2Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms. This ratio is called the common ratio (r).
nth Term Formula
aₙ = a₁ × r^(n-1)
aₙ
nth term
a₁
first term
r
common ratio
Example 1: Find the 5th term of 2, 6, 18, 54, ...
a₁ = 2, r = 6/2 = 3
a₅ = 2 × 3^(5-1)
a₅ = 2 × 3⁴
a₅ = 2 × 81
a₅ = 162
Example 2: Find a₁ if a₃ = 36 and r = 3
36 = a₁ × 3^(3-1)
36 = a₁ × 9
a₁ = 4
Finding Common Ratio
r = a₂/a₁ = a₃/a₂ = ... (any term divided by the previous)
3Harmonic Sequences
A harmonic sequence is the reciprocals of an arithmetic sequence. If a, b, c is arithmetic, then 1/a, 1/b, 1/c is harmonic.
Example: Harmonic Sequence
Arithmetic: 1, 2, 3, 4, 5, ...
Harmonic: 1, 1/2, 1/3, 1/4, 1/5, ...
Finding Terms in Harmonic Sequence
- Take reciprocals to get arithmetic sequence
- Find the term using arithmetic formula
- Take reciprocal of the answer
Example: 6th term of 1, 1/3, 1/5, 1/7, ...
Arithmetic: 1, 3, 5, 7, ... (d = 2)
a₆ = 1 + (6-1)(2) = 11
Harmonic a₆ = 1/11
4Fibonacci Sequence
In a Fibonacci sequence, each term is the sum of the two preceding terms. It starts with 1, 1 (or sometimes 0, 1).
Fibonacci Rule
aₙ = aₙ₋₁ + aₙ₋₂
Each term = Previous term + Term before that
The Fibonacci Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, ...
Fibonacci in Nature
- • Flower petals (lilies=3, buttercups=5, daisies=34)
- • Spiral patterns in shells and sunflowers
- • Branching of trees
- • Pinecone scales
5Sum Formulas
Arithmetic Series Sum
Sₙ = n/2 × (a₁ + aₙ)
When you know first and last terms
Sₙ = n/2 × [2a₁ + (n-1)d]
When you know first term and d
Example: Sum of first 20 terms of 2, 5, 8, 11, ...
a₁ = 2, d = 3, n = 20
S₂₀ = 20/2 × [2(2) + (20-1)(3)]
S₂₀ = 10 × [4 + 57]
S₂₀ = 10 × 61
S₂₀ = 610
Geometric Series Sum
Sₙ = a₁(1 - rⁿ)/(1 - r)
When r ≠ 1
S∞ = a₁/(1 - r)
Infinite sum when |r| < 1
Example: Sum of first 6 terms of 3, 6, 12, 24, ...
a₁ = 3, r = 2, n = 6
S₆ = 3(1 - 2⁶)/(1 - 2)
S₆ = 3(1 - 64)/(-1)
S₆ = 3(-63)/(-1)
S₆ = 189