Compound Interest as a Series
Discover the powerful connection between compound interest and geometric sequences, and learn to calculate future value using series formulas.
Compound Interest and Geometric Sequences
When money is invested at compound interest, the balance at the end of each compounding period forms a geometric sequence. If you invest a principal P at an interest rate r per period, the balance after n periods is:
A = P(1 + r)n
This is the nth term of a geometric sequence with first term P and common ratio (1 + r).
Each successive balance is obtained by multiplying the previous balance by the common ratio (1 + r). For example, at 5% per annum:
Visual: $1,000 at 5% p.a. Compound Interest
| Year | Balance | Calculation |
|---|---|---|
| 0 | $1,000.00 | P |
| 1 | $1,050.00 | 1000 x 1.05 |
| 2 | $1,102.50 | 1000 x 1.052 |
| 3 | $1,157.63 | 1000 x 1.053 |
| n | $1,000 x 1.05n | P(1 + r)n |
Geometric Series and Future Value
When regular contributions are made (e.g., depositing $M at the end of each period), the total future value involves summing a geometric series. Each deposit grows for a different number of periods:
The first deposit grows for n - 1 periods, the second for n - 2 periods, and so on:
FV = M(1+r)n-1 + M(1+r)n-2 + ... + M(1+r) + M
This is a geometric series with first term M, common ratio (1 + r), and n terms. Using the sum formula:
FV = M x (1 + r)n - 1r
This formula is the future value of an ordinary annuity. It tells us the total accumulated value of all regular payments plus the compound interest earned on each payment.
Simple vs Compound Interest Growth
Simple interest grows linearly (arithmetic sequence), while compound interest grows exponentially (geometric sequence). Over time, compound interest dramatically outpaces simple interest.
Visual: $1,000 over 10 years at 8% p.a.
The compound interest earns $359 more than simple interest over 10 years. The gap widens dramatically over longer periods.
Key Vocabulary
Geometric Sequence
A sequence where each term is obtained by multiplying the previous term by a constant called the common ratio.
Common Ratio
The fixed multiplier between consecutive terms. For compound interest at rate r, the common ratio is (1 + r).
Future Value (FV)
The total accumulated value of an investment or series of payments at a specified date in the future, including interest earned.
Geometric Series
The sum of a geometric sequence. Sn = a(rn - 1) / (r - 1) where a is the first term and r is the common ratio.
Worked Examples
Find the balance after 5 years if $2,000 is invested at 6% p.a. compounded annually.
Step 1: Identify values: P = $2,000, r = 0.06, n = 5.
Step 2: Apply the compound interest formula: A = P(1 + r)n = 2000(1.06)5.
Step 3: Calculate: A = 2000 x 1.338226... = $2,676.45.
Answer: The balance after 5 years is $2,676.45.
$500 is deposited at the end of each year into an account earning 4% p.a. Find the total after 10 years.
Step 1: Identify values: M = $500, r = 0.04, n = 10.
Step 2: Apply the future value formula: FV = M x [(1 + r)n - 1] / r.
Step 3: FV = 500 x [(1.04)10 - 1] / 0.04 = 500 x [1.480244 - 1] / 0.04.
Step 4: FV = 500 x 0.480244 / 0.04 = 500 x 12.00611 = $6,003.05.
Answer: The total accumulated value is $6,003.05.
$10,000 is invested at 3% p.a. compounded quarterly. What is the balance after 4 years?
Step 1: Identify values: P = $10,000, annual rate = 3%, compounding quarterly so r = 0.03/4 = 0.0075, n = 4 x 4 = 16 periods.
Step 2: A = P(1 + r)n = 10000(1.0075)16.
Step 3: A = 10000 x 1.12699... = $11,269.93.
Answer: The balance after 4 years is $11,269.93.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
$5,000 is invested at 8% p.a. compounded annually. What is the balance after 3 years?
Question 2
An investment of $1,000 at 5% p.a. compounded annually forms a geometric sequence. What is the common ratio?
Question 3
$200 is deposited at the end of each month into an account earning 6% p.a. compounded monthly. What is the monthly interest rate?
Question 4
Which formula gives the future value of regular payments of $M at the end of each period for n periods at rate r per period?
Question 5
$1,000 is deposited at the end of each year into an account earning 10% p.a. for 5 years. What is the future value?
Key Concepts Summary
- ● Compound interest balances form a geometric sequence with common ratio (1 + r).
- ● The compound interest formula is A = P(1 + r)n.
- ● Regular contributions create a geometric series; the future value is FV = M x [(1 + r)n - 1] / r.
- ● When interest is compounded more frequently, divide the annual rate by the number of periods and multiply the number of years by the frequency.
- ● Compound interest grows exponentially, outpacing simple (linear) interest over time.