Learning Objectives
Distinguish additive growth from multiplicative growth.
Select a term, sum, or logarithmic calculation to match a question.
Explain indexing, units, and whole-stage decisions in a model.
Prerequisites: the three sequence, series, and exponential-model lessons in this section. This review consolidates shared foundations; track-specific applications can build on them in later sections.
1. Identify the Quantity Requested
Request | Suitable expression |
|---|---|
Arithmetic term at position n | u₁ + (n − 1)d |
Geometric term at position n | u₁rⁿ⁻¹ |
Total of n arithmetic terms | n(u₁ + uₙ)/2 |
Total of n geometric terms, r ≠ 1 | u₁(1 − rⁿ)/(1 − r) |
Time to reach B in Abᵗ | ln(B/A)/ln b, when A,B > 0 and b > 0, b ≠ 1 |
Before inserting numbers, say what the result will represent: one value, a cumulative total, or an elapsed time. This sentence helps you choose the formula and later attach appropriate units.
2. Compare Two Models
Worked Example: Same Start, Different Change
Model A starts at 100 and adds 20 each stage. Model B starts at 100 and increases by 20% each stage. Let t = 0 be the start.
Their formulas are A(t) = 100 + 20t and B(t) = 100(1.2)ᵗ.
At t = 1 both equal 120. At t = 2, A(2) = 140 but B(2) = 144. Agreement at the first two observations does not make the models identical.
After three stages, A(3) = 160 and B(3) = 172.8. The difference comes from applying the percentage to a changing base.
3. Check the Meaning of an Answer
An exact logarithmic expression may give a fractional time. If the question asks for completed whole stages, test nearby integers. Likewise, solving for a sequence index must produce an allowed integer before you can claim that the value occurs as a term.
Common Mistakes
Do not round each stage of an exact model unless the situation requires it. Early rounding can change a later threshold decision.
Do not assume a finite set of observations proves that a model applies forever.
Check whether the initial quantity is indexed by 0 or 1 before counting changes.
4. Mixed Practice
1. An arithmetic sequence has u₁ = 9 and u₆ = 29. Find d and u₁₅.
2. Find the sum of the first 15 terms of that sequence.
3. A geometric sequence begins 81, 27, 9, … . Find its fifth term and the sum of its first five terms.
4. A quantity starts at 400 and loses 15% per stage. Write Q(t), where t counts completed stages from 0, and calculate Q(2).
5. Solve 50(1.1)ᵗ = 100 exactly. Decide the first whole stage at which the model reaches at least 100.
6. Explain why ln(4 + 5) is not equal to ln 4 + ln 5.
Worked Solutions
1. There are five steps from u₁ to u₆, so 29 = 9 + 5d gives d = 4. Then u₁₅ = 9 + 14(4) = 65.
2. S₁₅ = 15(9 + 65)/2 = 555.
3. r = 1/3. The fifth term is 81(1/3)⁴ = 1. The first five terms are 81, 27, 9, 3, 1, whose sum is 121.
4. Q(t) = 400(0.85)ᵗ, so Q(2) = 289. The retained fraction is 85%, not 15%.
5. The exact solution is t = ln 2/ln 1.1, approximately 7.27. At stage 7 the quantity is approximately 97.44, and at stage 8 it is approximately 107.18. The first qualifying whole stage is 8.
6. The left expression equals ln 9. By the product law, the right expression equals ln 20. Since 9 and 20 differ and ln is one-to-one on positive inputs, the expressions differ.
5. Summary
- A constant difference and a constant ratio describe different models.
- Name the requested quantity before selecting a formula.
- A correct calculation still needs correct units, indexing, and interpretation.
- Use substitution or a short list of terms to check answers when feasible.