Compound interest is a geometric sequence
Big idea: With compound interest, each year your balance is multiplied by the same number. Multiply by the same number every step — that is the definition of a geometric sequence.
Suppose you put $1 000 in a bank account that pays 5% interest per year. Each year the bank adds 5% of whatever is currently in the account — not 5% of the original deposit. So the balance grows like this:
What makes this geometric?
- u₁ = 1 000 (the starting balance)
- r = 1.05 (you keep 100% and add 5%, so multiply by 1 + 5/100 = 1.05)
- uₙ = 1 000 × 1.05ⁿ⁻¹ (standard geometric term formula)
Simple vs compound interest: Simple interest adds the same fixed amount each year (arithmetic — same difference). Compound interest multiplies by the same factor each year (geometric — same ratio). IB exams almost always mean compound interest unless they say otherwise.
Worked example — spot the geometric sequence
A savings account starts with $2 000 and earns 3% compound interest per year. Write the first four balances as a sequence and state u₁ and r.
Step by step
- Find r: adding 3% means you multiply by 1 + 3/100.
- Write the first four terms using uₙ = u₁ × rⁿ⁻¹.
- State u₁ and r.
Final answer
u₁ = 2 000, r = 1.03
The IB compound interest formula
Big idea: The IB formula handles any compounding frequency — yearly, monthly, quarterly, or daily — with a single equation. Learn each letter and you can answer any compound interest question.
What each letter means
- FV = future value (what you end up with)
- PV = present value (what you start with)
- r = annual interest rate as a percentage (e.g. 5, not 0.05)
- k = number of compounding periods per year (k = 1 yearly, k = 4 quarterly, k = 12 monthly)
- n = number of years
Common values of k
| Compounding | k |
|---|---|
| Yearly | 1 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
Worked example 1 — yearly compounding
Find the value after 6 years of $5 000 invested at 4% per year, compounded yearly.
Step by step
- Write down PV, r, k, and n.
- Substitute into the formula.
- Simplify inside the bracket.
- Calculate.
Final answer
FV ≈ $6 326.60
Worked example 2 — monthly compounding
Find the value after 3 years of $2 000 invested at 6% per year, compounded monthly.
Step by step
- Write down PV, r, k, and n.
- Substitute into the formula.
- Simplify: r/(100k) = 6/1200 = 0.005, and kn = 36.
- Calculate.
Final answer
FV ≈ $2 393.40
r is a percentage — not a decimal: In this formula, enter r = 4 (not 0.04). The 100 in the denominator does the conversion for you. This is different from many textbooks — watch out.
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Using the GDC TVM solver
Big idea: Your GDC has a built-in finance app called the TVM solver (Time Value of Money). It solves compound interest problems in seconds — and IB Paper 2 expects you to use it.
What each input means
- N = total number of compounding periods = k × n
- I% = annual interest rate as a percentage (e.g. 4, not 0.04)
- PV = present value (enter as negative if money is going out)
- PMT = payment per period (enter 0 for a lump sum — no regular payments)
- FV = future value (what you want to find, or enter it to find N)
- P/Y = payments per year (usually 1 for lump sums)
- C/Y = compounding periods per year = k
Sign convention: Money leaving your pocket is negative. Money coming in is positive. If you invest $5 000, enter PV = −5 000. The FV answer will be positive (money coming back to you).
Worked example — TVM solver
Use the GDC TVM solver to find the future value of $5 000 invested at 4% per year for 6 years, compounded yearly.
Step by step
- Open the Finance app and choose TVM Solver.
- Enter N = k × n = 1 × 6.
- Enter the annual interest rate.
- Enter PV as negative (money going out).
- Enter PMT = 0 (lump sum, no regular payments).
- Set P/Y and C/Y to k = 1 (yearly).
- Move cursor to FV and press ALPHA + ENTER (or SOLVE) to calculate.
Final answer
FV ≈ $6 326.60 — matches the formula answer from Section 2
When to use the TVM solver vs the formula: Use the formula when the question is straightforward and you want to show working. Use the TVM solver when you need to find N (how many years) or when the arithmetic gets messy — the solver does it instantly.
Exam-style compound interest questions
What IB asks: IB compound interest questions typically come in three flavours: find the future value, find how many years to reach a target, or compare two investments. This section walks through all three.
Type 1 — Find the future value
Worked example — find FV
Hamid invests $8 000 at 3.5% per year compound interest, compounded quarterly. Find the value of his investment after 5 years.
Step by step
- Write down PV, r, k, and n.
- Substitute into the formula.
- Simplify: r/(100k) = 3.5/400 = 0.00875, and kn = 20.
- Calculate.
Final answer
FV ≈ $9 525.52
Type 2 — Find how many years
Worked example — find n
Maria invests 4 000?
Step by step
- Set up the inequality.
- Divide both sides by 3000.
- Use the TVM solver: set PV = −3000, FV = 4000, I% = 5, C/Y = 1, PMT = 0. Solve for N.
- N = 6.02 means 6 full years is not quite enough — she needs 7 full years.
Final answer
7 full years
Type 3 — Compare two investments
Worked example — compare investments
Plan A: 10 000 at 3.8% per year compounded monthly for 10 years. Which gives more?
Step by step
- Calculate Plan A: k = 1, n = 10.
- Calculate Plan B: k = 12, n = 10, so kn = 120.
- Compare the two final values.
Final answer
Plan A gives $208.70 more after 10 years.
Exam traps
- Entering r as a decimal (0.04) instead of a percentage (4) — the formula already divides by 100
- Forgetting to multiply k × n for the exponent when compounding is not yearly
- Rounding too early — keep full precision until the final answer
- For 'how many full years' questions: always round up, even if the decimal is small
- Confusing simple interest and compound interest — compound uses multiplication, not addition