What is a Sequence?
The foundation you need first
Sequence
A function from the whole numbers (0, 1, 2, 3, …) to the real numbers. You can only plug in whole numbers.
Discrete, not continuous
When you graph a sequence, you get isolated dots — you cannot connect them into a line or curve.
Notation: aₙ or gₙ
The subscript n is the input (term number). a₁ means the 1st term, a₀ is the initial term.
📌 Example 1 — Given aₙ = 4n − 3, find a₁ and a₇.
1
Substitute n = 1: a₁ = 4(1) − 3 = 4 − 3 = 12
Substitute n = 7: a₇ = 4(7) − 3 = 28 − 3 = 25a₁ = 1 a₇ = 25
Arithmetic vs Geometric
The two types you need to know
📐 Arithmetic Sequence
Key Property
Common difference d
Successive terms
aₙ₊₁ − aₙ = d (constant)
Formula from a₀
aₙ = a₀ + dn
Formula from aₖ
aₙ = aₖ + d(n − k)
Behaves like
Linear function (discrete)
Example
aₙ = 3n + 1 (d = 3)
📈 Geometric Sequence
Key Property
Common ratio r
Successive terms
aₙ₊₁ / aₙ = r (constant)
Formula from g₀
gₙ = g₀ · rⁿ
Formula from gₖ
gₙ = gₖ · r^(n−k)
Behaves like
Exponential function (discrete)
Example
gₙ = 8·(1/2)ⁿ (r = 1/2)
Identifying Sequence Type
Examples 2 & 6 from the notes
How to identify the type
For arithmetic: subtract consecutive terms — if the difference is constant, it's arithmetic.
For geometric: divide consecutive terms — if the ratio is constant, it's geometric.
If neither is constant → it's neither.
📌 Example 2a — Is sₙ = n² − 3 arithmetic?
1
Compute terms: n=0→−3, n=1→−2, n=2→1, n=3→62
Differences: −2−(−3)=1, 1−(−2)=3, 6−1=5 — not constant❌ Not arithmetic — no common difference
📌 Example 2b — Is sₙ = 6 − 2n arithmetic?
1
Compute terms: n=0→6, n=1→4, n=2→2, n=3→02
Differences: 4−6=−2, 2−4=−2, 0−2=−2 — constant!✅ Arithmetic — d = −2
📌 Example 2c — Is −7, −2, 3, 8, 13, … arithmetic?
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Differences: −2−(−7)=5, 3−(−2)=5, 8−3=5, 13−8=5✅ Arithmetic — d = 5
📌 Example 6b — Is sₙ = 4(2)^(n−1) geometric?
1
Compute terms: n=1→4, n=2→8, n=3→16, n=4→322
Ratios: 8/4=2, 16/8=2, 32/16=2 — constant!✅ Geometric — r = 2
📌 Example 6d — Is 16, −8, 4, −2, 1, … geometric?
1
Ratios: −8/16 = −1/2, 4/(−8) = −1/2, −2/4 = −1/2✅ Geometric — r = −1/2
Arithmetic Sequence Examples
Using the formula aₙ = aₖ + d(n − k)
The key formula
If you know any term aₖ and the common difference d, you can find any other term:
aₙ = aₖ + d(n − k)
If you know two terms but not d, use the formula twice to solve for d first.
📌 Example 3 — Arithmetic sequence with a₃ = 8 and d = −3. Find aₙ and a₁₂.
1
Use aₙ = aₖ + d(n − k) with k=3: aₙ = 8 + (−3)(n − 3)2
Expand: aₙ = 8 − 3n + 9 = 17 − 3n3
Find a₁₂: a₁₂ = 17 − 3(12) = 17 − 36 = −19aₙ = 17 − 3n a₁₂ = −19
📌 Example 4 — Arithmetic sequence with a₂ = 7 and a₆ = 9. Find aₙ and a₂₄.
1
Use a₆ = a₂ + d(6 − 2): 9 = 7 + 4d → 4d = 2 → d = 1/22
Build formula: aₙ = a₆ + (1/2)(n − 6) = 9 + (n−6)/23
Find a₂₄: a₂₄ = 9 + (1/2)(24 − 6) = 9 + 9 = 18d = 1/2 a₂₄ = 18
📌 Example 5 — Graph shows a₀ = 8, a₁ = 6. Find aₙ and a₁₇.
1
Find d: a₁ − a₀ = 6 − 8 = −22
Use aₙ = a₀ + dn: aₙ = 8 + (−2)n = 8 − 2n3
Find a₁₇: a₁₇ = 8 − 2(17) = 8 − 34 = −26aₙ = 8 − 2n a₁₇ = −26
Geometric Sequence Examples
Using the formula gₙ = gₖ · r^(n−k)
The key formula
If you know any term gₖ and the common ratio r:
gₙ = gₖ · r^(n−k)
To find r from a table or graph: r = g₁ / g₀ (any consecutive pair works).
📌 Example 7 — Geometric sequence with g₁ = 12 and r = 2. Find gₙ and g₄.
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Use gₙ = gₖ · r^(n−k) with k=1: gₙ = 12 · 2^(n−1)2
Find g₄: g₄ = 12 · 2^(4−1) = 12 · 2³ = 12 · 8 = 96gₙ = 12 · 2^(n−1) g₄ = 96
📌 Example 8 — Table shows g₀ = 8, g₁ = 4. Find gₙ and g₁₀.
1
Find r: g₁/g₀ = 4/8 = 1/22
Use gₙ = g₀ · rⁿ: gₙ = 8 · (1/2)ⁿ3
Find g₁₀: g₁₀ = 8 · (1/2)¹⁰ = 8/1024 = 1/128gₙ = 8 · (1/2)ⁿ g₁₀ = 1/128
Quick Reference
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📐 Arithmetic Formulas
aₙ = a₀ + dn
From initial term a₀
aₙ = aₖ + d(n − k)
From any known term aₖ
d = aₙ − aₙ₋₁
Common difference = subtract consecutive terms
📈 Geometric Formulas
gₙ = g₀ · rⁿ
From initial term g₀
gₙ = gₖ · r^(n−k)
From any known term gₖ
r = gₙ / gₙ₋₁
Common ratio = divide consecutive terms
⚠️ Common Mistakes
❌ Connecting the dots on a sequence graph
Sequences are discrete — only whole number inputs are valid. Never draw a line through the points.
❌ Confusing d (difference) with r (ratio)
Arithmetic → subtract to check. Geometric → divide to check. Don't mix them up!
❌ Using wrong k in aₙ = aₖ + d(n − k)
Whatever term you're starting from is k. If you know a₃, then k=3. If you know a₀, then k=0.