The big idea: Money, populations and values that change by the same percentage each period form a geometric sequence.
The common ratio is r = 1 + rate for growth, or r = 1 โ rate for decay.
For example, 6% growth โ r = 1.06; 15% decay โ r = 0.85.
Translate the words
- "grows / increases by x%" โ r = 1 + x/100.
- "falls / depreciates by x%" โ r = 1 โ x/100.
- "doubles / triples" โ solve rโฟ = 2 or 3.
- "value after t periods" โ start ร rแต.
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Compound interest is geometric: A classic Paper 2 question asks how long money takes to reach a target โ model the balance as start ร rโฟ and solve for n (logs or the TVM solver).
IB-style question โ when does it double?
$2000 is invested at 6% per year, compounded annually.
After how many whole years does it first exceed $4000?
Step by step
- As a geometric sequence, the balance after n years is start ร rโฟ.
- Divide by 2000 โ it doubles.
- Solve (logs or the GDC) and round up.
Final answer
12 years.
GDC walkthrough
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Two ways, same answer: On Paper 2 you can use the TVM solver (above) or the geometric way (solve rโฟ = 2 with logs).
Both give n โ 11.9 โ 12 years. Always round up for "how long until".
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Decay multiplies by less than 1: When something shrinks by the same percentage each period (depreciation, cooling), the ratio is r = 1 โ rate (between 0 and 1) โ then model and solve exactly like growth.
IB-style question โ depreciation
A machine worth $20 000 loses 15% of its value each year.
(a) Find its value after 4 years. (b) After how many whole years is it first worth less than $8000?
Step by step
- Decay ratio.
- (a) Value after 4 years = start ร rโด.
- (b) Set below 8000 and solve, then round up.
Final answer
(a) โ $10 440. (b) After 6 years.
Rounding for decay: Still round n up for "how many years until below $X".
Because the value keeps a fixed percentage each year, it shrinks but never quite reaches zero.