Lesson 10 of 12 · Algebra Foundations
Sequences, Series & Recursion
Arithmetic and geometric sequences each have exactly two formulas worth memorizing — one for a term, one for a sum. Recursively defined sequences have no shortcut formula at all; the skill there is just careful, systematic computation, term by term.
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Learning objectives
- Find any term of an arithmetic or geometric sequence, and find missing parameters (d, r, or a₁) from given terms.
- Find the sum of a finite arithmetic or geometric series, and the sum of an infinite geometric series when it converges.
- Compute terms of a recursively defined sequence systematically.
- Recognize when Sₙ − Sₙ₋₁ reveals a hidden pattern in a sequence's individual terms.
The core idea
An arithmetic sequence adds the same amount each step; a geometric sequence multiplies by the same amount each step. Both have a direct formula for any term and a direct formula for the sum of the first n terms — no need to add terms one at a time once you know a₁ and d (or r). A recursively defined sequence, by contrast, only tells you how to get the next term from the previous ones — there's no shortcut without more context, so you compute term by term.
Arithmetic sequence and series
aₙ = a₁ + (n−1)d
Sₙ = (n/2)(a₁ + aₙ)
Geometric sequence and series
aₙ = a₁ · rⁿ⁻¹
Sₙ = a₁(rⁿ−1)/(r−1) for finite sums; S∞ = a₁/(1−r) for the infinite sum, only when |r| < 1
Extracting a term from partial sums
aₙ = Sₙ − Sₙ₋₁
Works for any sequence, not just arithmetic or geometric ones — useful whenever a problem gives you a formula for the running sum instead of the sequence itself.
Worked example 1 — A term of an arithmetic sequence
Problem
Find the 15th term of the sequence 3, 7, 11, 15, ...
Key insight
Identify a₁ = 3 and d = 4, then apply the term formula directly.
Solution
a₁₅ = 3 + 14(4) = 59.
Takeaway
Never list out 15 terms by hand — the formula gets you there in one step once a₁ and d are identified.
Worked example 2 — Sum of an arithmetic series
Problem
Find the sum of the first 20 terms of 5, 8, 11, 14, ...
Key insight
Find a₂₀ first, then apply the sum formula — you need both endpoints.
Solution
a₂₀ = 5 + 19(3) = 62, so S₂₀ = (20/2)(5+62) = 670.
Takeaway
The sum formula needs the last term, not just the first — always find aₙ before applying it.
Worked example 3 — A term of a geometric sequence
Problem
Find the 8th term of 2, 6, 18, 54, ...
Key insight
Identify a₁ = 2 and r = 3, then apply the term formula directly.
Solution
a₈ = 2·3⁷ = 2(2187) = 4374.
Takeaway
Geometric terms grow fast — a formula is essential here, not optional, once n gets past a handful.
Worked example 4 — Computing a recursive sequence
Problem
A sequence satisfies a₁=1, a₂=1, and aₙ=aₙ₋₁+aₙ₋₂ for n≥3 (the Fibonacci sequence). Find a₁₀.
Key insight
There's no direct formula available here — compute each term in order.
Solution
1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Takeaway
Stay organized — write out each term with its index as you go, so a single arithmetic slip doesn't propagate silently through the rest of the sequence.
Strategy notes
Identify a₁, d (or r) before doing anything else
Every arithmetic or geometric sequence problem starts the same way: pin down the first term and the common difference or ratio. Everything else follows mechanically from there.
For a recursive sequence, just compute — don't search for a shortcut
Unless a problem explicitly hints at a closed-form pattern, the fastest path through a recursion problem is careful, organized term-by-term computation.
Common mistakes
Off-by-one errors in (n−1)
aₙ = a₁ + (n−1)d, not a₁ + nd. This single off-by-one error is the most common mistake in the entire topic — double-check it whenever a problem gives an unusual starting index.
Using the infinite sum formula when |r| ≥ 1
S∞ = a₁/(1−r) only converges to a finite value when |r| < 1. Applying it when |r| ≥ 1 produces a meaningless answer — the series simply diverges in that case.
Losing track of indices in a recursion
When computing term by term, it's easy to accidentally use the wrong pair of previous terms, especially past a₅ or so. Label every term with its index as you write it down.
Practice
20 questions across four difficulty tiers, each with a single numeric answer. Up to three tries per question before the solution is shown.