Math 20-1 · Sequences and series
Sequences and series
A sequence is a list; a series is what you get when you add the list up. Two kinds are worth knowing — the one that adds the same thing each step, and the one that multiplies — and each has a formula for the nth term and a formula for the sum.
- 1. The words, first
- 2. Arithmetic sequences and series
- 3. Geometric sequences and series
- 4. Infinite geometric series
- 5. What costs marks
The words, first
The idea: Most errors in this unit are a term formula used where a sum formula belonged, or an off-by-one in n − 1.
| Word | What it means |
|---|---|
| Sequence | An ordered list of numbers: 3, 7, 11, 15, … |
| Series | The sum of the terms of a sequence: 3 + 7 + 11 + 15. |
| Term / tn | One entry in the list. t₁ is the first, tₙ the general or nth term. |
| Common difference d | What is added each step in an arithmetic sequence. Found by subtracting any term from the one after it. |
| Common ratio r | What is multiplied each step in a geometric sequence. Found by dividing any term by the one before it. |
| Arithmetic | Constant difference. Plotted, the terms sit on a straight line. |
| Geometric | Constant ratio. Plotted, the terms curve like an exponential. |
| Sn | The sum of the first n terms — a partial sum. |
| Convergent series | An infinite series whose partial sums approach a fixed number. Only geometric series with −1 < r < 1 do this. |
| Divergent series | One whose partial sums do not settle. Every arithmetic series with d ≠ 0 diverges. |
Arithmetic sequences and series
The idea: Add d each step, so the nth term is the first term plus (n − 1) of them. There are n − 1 gaps between n terms, not n.
tₙ = t₁ + (n − 1)d · Sₙ = n/2 [2t₁ + (n − 1)d] · or Sₙ = n/2 (t₁ + tₙ) when you know the last term
Finding a term. t₁ = 5, d = 3: t₁₀ = 5 + 9(3) = 32. Nine steps, not ten.
Finding d from two terms. t₁ = 4 and t₈ = 32: 32 = 4 + 7d, so d = 4. Seven gaps between the first and eighth terms.
Summing. For 3, 7, 11, … the first 20 terms give S₂₀ = 10[6 + 19(4)] = 10(82) = 820. The second formula makes the reason clear: the average term is (3 + 79)/2 = 41, and 20 × 41 = 820.
Solving for n. If t₁ = 2, d = 3 and Sₙ = 155, then n(3n + 1)/2 = 155, so 3n² + n − 310 = 0 and n = 10. The negative root is discarded — a number of terms is a positive whole number.
Geometric sequences and series
The idea: Multiply by r each step. The exponent is n − 1 for the same reason as before, and the sum formula is a subtraction trick rather than something to memorise blindly.
tₙ = t₁ rn−1 · Sₙ = t₁(rⁿ − 1)/(r − 1), r ≠ 1
Finding a term. t₁ = 2, r = 3: t₅ = 2 × 3⁴ = 162. Using 3⁵ gives 486, which is the sixth term.
Summing. 2, 6, 18, … for six terms: S₆ = 2(3⁶ − 1)/(3 − 1) = 728. The sixth term alone is 486, so a total of 728 is the right size.
Geometric mean. If 5, x, 45 are consecutive, then x/5 = 45/x, so x² = 225 and x = ±15. Both work — one sequence has ratio 3 and the other −3 — so giving only 15 loses half the answer.
Infinite geometric series
The idea: Adding forever can still give a finite total, but only when each term is a genuine fraction of the one before it.
S∞ = t₁/(1 − r), and only when −1 < r < 1
Worked example. 12 + 6 + 3 + 1.5 + … has r = ½, so S = 12/(1 − 0.5) = 24. The terms shrink fast enough that the total never passes 24.
Why the condition. At r = 1 the terms never shrink, so the sum grows without limit. At r = −1 they flip between two values forever and the partial sums never settle. Both endpoints are excluded, and anything outside diverges.
Repeating decimals. 0.3̄ = 0.3 + 0.03 + 0.003 + … is geometric with t₁ = 0.3 and r = 0.1, so S = 0.3/0.9 = ⅓. That is what a repeating decimal is.
What costs marks
The idea: Two of these are arithmetic slips; two are reading the question.
- Using n instead of n − 1. The first term takes no steps to reach.
- Mixing up a term and a sum. “The tenth term” and “the sum of ten terms” are different questions with different formulas.
- Applying S∞ when |r| ≥ 1. Check the ratio before using the formula; the answer is “it diverges”.
- Giving only the positive geometric mean. x² = 225 has two solutions and both give a valid sequence.