Learning Objectives
Recognize arithmetic and geometric sequences from their terms.
Write an nth-term formula using a clearly stated starting index.
Solve for a missing term or an index and interpret it in context.
Scope: shared mathematical foundations for IB DP learners, not a complete AA/AI or SL/HL syllabus map. Prerequisites: substitution, fractions, and exponent rules. In this section, the first term is u₁ and n is a positive integer.
1. A Sequence Has Discrete Inputs
A sequence is an ordered list of terms. We write uₙ for the term at position n. The subscript identifies a position, not a multiplication. An explicit formula gives uₙ directly; a recurrence tells you how to calculate a term from earlier terms.
2. Arithmetic Sequences Add a Fixed Difference
An arithmetic sequence has constant difference d = uₙ₊₁ − uₙ. Starting with u₁, its nth term is uₙ = u₁ + (n − 1)d. There are n − 1 steps from the first term to the nth term, which explains the exponent-free factor n − 1.
Worked Example: Find a Term and an Index
For 7, 11, 15, 19, …, the common difference is 4. Thus uₙ = 7 + 4(n − 1).
The tenth term is u₁₀ = 7 + 4(9) = 43.
To find the position of 55, solve 7 + 4(n − 1) = 55. This gives n − 1 = 12 and n = 13. The positive integer answer is a valid position.
An arithmetic sequence may increase, decrease, or stay constant. The sign of d determines which: positive, negative, or zero. If solving for n gives a noninteger, the requested value is not a term under this indexing convention.
3. Geometric Sequences Multiply by a Fixed Ratio
A geometric sequence follows uₙ₊₁ = ruₙ, where r is constant. Its explicit formula is uₙ = u₁rⁿ⁻¹. When the preceding term is nonzero, divide adjacent terms to calculate r. Check more than one pair to see whether a single ratio fits the given data.
Worked Example: A Negative Ratio
For 3, −6, 12, −24, …, each term is multiplied by −2. Hence uₙ = 3(−2)ⁿ⁻¹.
The fifth term is 3(−2)⁴ = 48. The signs alternate because the ratio is negative.
Parentheses matter: (−2)⁴ is positive, whereas −2⁴ means the negative of 2⁴.
Sequence | Test | Model |
|---|---|---|
5, 8, 11, 14 | Difference 3 | uₙ = 5 + 3(n − 1) |
5, 10, 20, 40 | Ratio 2 | uₙ = 5·2ⁿ⁻¹ |
1, 4, 9, 16 | Neither test is constant | Neither of these two models |
4. Model a Repeated Process
A tank containing 200 litres loses 10% of its current contents at each stage. If u₁ = 200 is the initial amount, each stage retains 90%, so uₙ = 200(0.9)ⁿ⁻¹. After three losses, the relevant term is u₄ = 145.8 litres. If instead the initial amount were named u₀, the same process would use uₙ = 200(0.9)ⁿ. State the convention to avoid an off-by-one error.
Common Mistakes
A constant percentage change gives a geometric model, not a constant additive difference.
Do not use n rather than n − 1 when the initial amount is u₁.
Several initial terms suggest a model but do not prove it describes an unobserved process indefinitely.
5. Practice
1. Find u₈ for the arithmetic sequence with u₁ = 20 and d = −3.
2. Find u₆ for 64, 32, 16, … and state r.
3. Is 50 a term of 4, 10, 16, …? Show how you decide.
Worked Solutions
1. u₈ = 20 + 7(−3) = −1.
2. r = 1/2 and u₆ = 64(1/2)⁵ = 2.
3. Solve 4 + 6(n − 1) = 50 to obtain n = 26/3. This is not an integer, so 50 is not a term.
6. Summary
- Arithmetic sequences add a constant; geometric sequences multiply by a constant.
- State the starting index before choosing the nth-term formula.
- Check that an index is an allowed integer and that the model fits the context.