Chapter 3 · Test 1
Sequences, Series & Limits for Economics
Sequence limits, geometric series, convergence, and present value.
What you will learn
- Find a term of a geometric sequence
- Discount a future payment
- Sum a convergent geometric series
Worked example
Find a term of a geometric sequence: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Find a term of a geometric sequence
A geometric sequence has first term a₁ = 3 and common ratio r = 2. Find a₅.
aₙ = a₁rⁿ⁻¹
In plain English: The nth term of a geometric sequence equals the first term multiplied by the common ratio once for each step after the first.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- a₁
- the first termHow to read: a sub one
- r
- the Common ratio: The fixed multiplier that takes one term of a geometric sequence to the next.How to read: r
- aₙ
- the nth termHow to read: a sub n
- n
- the term numberHow to read: n
Solution
- Use aₙ = a₁rⁿ⁻¹.
- Substitute n = 5, a₁ = 3, and r = 2: a₅ = 3(2)⁴.
- Compute 2⁴ = 16.
- Multiply: a₅ = 3(16) = 48.
Logic and solving tips
- The exponent is n − 1 because the first term contains zero applications of the common ratio.
- Count how many times the ratio is applied: reaching term n requires n − 1 multiplications by r.
Where economists use it
A forecaster compounds a repeated inflation or growth rate to estimate a future price level, debt balance, population, or investment value.
Key idea: The exponent is n − 1 because the first term contains zero applications of the common ratio.
Applied case study
A price level growing at 2 percent per year
Scenario
Economists use geometric sequences for repeated proportional growth. The FOMC states a 2 percent longer-run PCE inflation goal; assume only for this exercise that a basket's price rises exactly 2 percent each year.
Problem
A basket costs $100 today. What would it cost after five years of constant 2 percent annual inflation?
Pₜ = P₀(1 + π)ᵗ = 100(1.02)⁵
In plain English: The future price level equals its starting value multiplied by the same proportional growth factor once per period.
Worked solution
- Convert 2 percent to π = 0.02.
- The annual common ratio is 1 + π = 1.02.
- Apply it for five years: P₅ = 100(1.02)⁵.
- Calculate: P₅ = 100(1.1040808) ≈ $110.41.
Economic interpretation: Under the constant-rate assumption, the basket becomes about 10.41 percent more expensive over five years because growth compounds.
Source: Federal Reserve Board: 2025 Statement on Longer-Run Goals and Monetary Policy Strategy The 2 percent longer-run PCE inflation goal is sourced. Exact 2 percent annual inflation and the $100 basket are teaching assumptions, not a forecast.
Reasoning habits that help
- The exponent is n − 1 because the first term contains zero applications of the common ratio.
- Discounting reverses compounding: future dollars are divided by the accumulated interest factor.
- An infinite geometric series converges exactly when the common ratio has absolute value below one.
Where economists use these methods
- A forecaster compounds a repeated inflation or growth rate to estimate a future price level, debt balance, population, or investment value.
- A government compares an infrastructure project's upfront cost with future benefits by discounting every payment to today's value before deciding whether the project is worthwhile.
- A financial economist sums a perpetual stream of coupons, rents, or maintenance costs to value an asset or long-lived policy commitment today.
Chapter vocabulary
Study 29 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Sequence{a_n}_{n=1}^∞
A function whose domain is the positive integers, producing an ordered list of terms.
How to read: the sequence a sub n from n equals one to infinity
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed page 61Term of a sequencea_n
One value in a sequence, identified by its position or index.
How to read: a sub n
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed page 61Bounded abovea_n ≤ M for every n
A sequence is bounded above when some finite number is at least as large as every term.
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed pages 62-63Bounded belowm ≤ a_n for every n
A sequence is bounded below when some finite number is no greater than every term.
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed pages 62-63Bounded sequence|a_n| ≤ M for some M and every n
A sequence that is bounded both above and below.
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed pages 62-63Unbounded sequence
A sequence for which no finite interval contains every term.
Textbook reference: Chapter 3, Section 3.1 Definition of a Sequence, printed pages 62-63Limit of a sequencelim_{n→∞} a_n = a
A number that the sequence's terms eventually approach arbitrarily closely and remain close to.
How to read: the limit of a sub n as n approaches infinity equals a
Textbook reference: Chapter 3, Section 3.2 Limit of a Sequence, printed page 65Convergent sequence
A sequence that has a finite limit.
Textbook reference: Chapter 3, Section 3.2 Limit of a Sequence, printed page 65Divergent sequence
A sequence that does not converge to a finite limit.
Textbook reference: Chapter 3, Section 3.2 Limit of a Sequence, printed pages 65-66Definite divergencea_n → +∞ or a_n → -∞
Divergence in which the terms eventually exceed every finite bound in one direction, approaching positive or negative infinity.
Textbook reference: Chapter 3, Section 3.2 Limit of a Sequence, printed page 66Discrete compoundingV_t = V_0(1 + r)^t
Interest accumulation at distinct intervals, with each interval's interest added to the balance for later intervals.
Textbook reference: Chapter 3, Section 3.3 Present-Value Calculations, printed pages 69-70Discount factor1/(1 + r)^t
The multiplier that converts a future amount to its value at an earlier date.
Textbook reference: Chapter 3, Section 3.3 Present-Value Calculations, printed pages 69-70Present valuePV = V_t/(1 + r)^t
The value today of a future payment or stream after discounting at a stated rate.
Textbook reference: Chapter 3, Section 3.3 Present-Value Calculations, printed page 70Net present valueNPV = PV(benefits) - PV(costs)
The present value of benefits or receipts minus the present value of costs or payments.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 91-92Continuous compoundingV(t) = V(0)e^{rt}
The limiting form of compounding as interest is credited at ever shorter intervals.
Textbook reference: Chapter 3, Section 3.3 Present-Value Calculations, printed page 72Internal rate of returnNPV(r) = 0
The discount rate that makes a project's net present value equal to zero.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 92-93Monotonically increasing sequencea_{n+1} ≥ a_n
A sequence in which each term is at least as large as the preceding term.
Textbook reference: Chapter 3, Section 3.4 Properties of Sequences, printed page 79Strictly increasing sequencea_{n+1} > a_n
A sequence in which each term is larger than the preceding term.
Textbook reference: Chapter 3, Section 3.4 Properties of Sequences, printed page 79Monotonically decreasing sequencea_{n+1} ≤ a_n
A sequence in which each term is no greater than the preceding term.
Textbook reference: Chapter 3, Section 3.4 Properties of Sequences, printed page 79Strictly decreasing sequencea_{n+1} < a_n
A sequence in which each term is smaller than the preceding term.
Textbook reference: Chapter 3, Section 3.4 Properties of Sequences, printed page 79SeriesΣ_{n=1}^∞ a_n
The sum of the terms of a sequence.
How to read: the sum from n equals one to infinity of a sub n
Textbook reference: Chapter 3, Section 3.5 Series, printed page 84Partial sumS_N = Σ_{n=1}^N a_n
The sum of the first N terms of a series.
Textbook reference: Chapter 3, Section 3.5 Series, printed page 84Convergent series
A series whose sequence of partial sums converges to a finite number.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 84-85Divergent series
A series whose sequence of partial sums does not converge to a finite number.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 84-85Harmonic seriesΣ_{n=1}^∞ 1/n
The divergent series formed by summing reciprocals of the positive integers.
Textbook reference: Chapter 3, Section 3.5 Series, printed page 86Geometric seriesΣ_{n=0}^∞ ar^n
A series whose successive terms are obtained by multiplying by a constant common ratio.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 87-88Common ratior = a_{n+1}/a_n
The fixed multiplier relating consecutive terms of a geometric sequence or series.
Textbook reference: Chapter 3, Section 3.5 Series, printed page 88Ratio testlim |a_{n+1}/a_n| < 1
A convergence test based on the limiting absolute ratio of successive terms, with a limit below one sufficient for convergence.
Textbook reference: Chapter 3, Section 3.5 Series, printed pages 89-90PerpetuityPV = V/r for equal end-of-period payments V and r > 0
A stream of payments that continues indefinitely and can be valued as a convergent discounted series.
Textbook reference: Chapter 3, Section 3.5 Series, printed page 89End-of-chapter problems
Practice all 7 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems