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Topic 2.1 · Definition
What is a sequence?
f : whole numbers → real numbers
What inputs are allowed?
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Topic 2.1 · Arithmetic Formula
What is the arithmetic sequence formula from the initial term a₀?
aₙ = a₀ + dn
d = common difference
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Topic 2.1 · Arithmetic Formula
What is the arithmetic sequence formula from any known term aₖ?
aₙ = aₖ + d(n − k)
Use this when a₀ is not given
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Topic 2.1 · Geometric Formula
What is the geometric sequence formula from the initial term g₀?
gₙ = g₀ · rⁿ
r = common ratio
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Topic 2.1 · Geometric Formula
What is the geometric sequence formula from any known term gₖ?
gₙ = gₖ · r^(n − k)
Use this when g₀ is not given
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Topic 2.1 · Common Difference
How do you find the common difference d of an arithmetic sequence?
Think: what operation between consecutive terms?
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Topic 2.1 · Arithmetic Example
Arithmetic sequence: a₃ = 8, d = −3. Find a₁₂.
aₙ = aₖ + d(n − k)
Start from a₃ with k = 3
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Topic 2.1 · Geometric Example
Geometric sequence: g₀ = 8, g₁ = 4. Find g₁₀.
gₙ = g₀ · rⁿ
Find r first from g₁/g₀
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✓ Answer — Arithmetic from a₀
aₙ = a₀ + dn
a₀ = initial term, d = common difference, n = term number.
Example: a₀=1, d=3 → aₙ = 1 + 3n
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✓ Answer — Sequence
aₙ or gₙ, whole number inputs only
Sequences are discrete — inputs are 0, 1, 2, 3, … only. Graph shows isolated dots, not a connected line.
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✓ Answer — Geometric from g₀
gₙ = g₀ · rⁿ
g₀ = initial term, r = common ratio.
Example: g₀=8, r=½ → gₙ = 8·(½)ⁿ
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✓ Answer — Arithmetic from aₖ
aₙ = aₖ + d(n − k)
Use when a₀ is unknown. k is the term number you know.
Example: a₃=8, d=−3 → a₁₂ = 8+(−3)(9) = −19
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✓ Answer — Common Difference
d = aₙ₊₁ − aₙ
Subtract any term from the next one.
Ratios → geometric. Differences → arithmetic.
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✓ Answer — Geometric from gₖ
gₙ = gₖ · r^(n − k)
Use when g₀ is unknown. k is the term number you know.
Example: g₁=12, r=2 → g₄ = 12·2³ = 96
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✓ Answer — Geometric Example
r = 1/2 → gₙ = 8·(½)ⁿ → g₁₀ = 1/128
r = g₁/g₀ = 4/8 = ½.
g₁₀ = 8·(½)¹⁰ = 8/1024 = 1/128
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✓ Answer — Arithmetic Example
aₙ = 17 − 3n → a₁₂ = −19
aₙ = 8 + (−3)(n−3) = 17 − 3n
a₁₂ = 17 − 36 = −19