Simple interest as equal increase
Big idea: Simple interest adds the same amount each year, so it can be modelled by an arithmetic sequence.
If the increase is the same every time, the balances form an arithmetic pattern.
Easy example:
- A savings account starts with 1000 and gains 50 each year.
- Balances: 1000, 1050, 1100, 1150, ...
- This is arithmetic because the common difference is 50.
Quick check
- Same increase each year -> arithmetic.
- The common difference is the interest added each year.
- This is not compound interest.
Use arithmetic formulas in context
In context questions, first decide what the terms mean. Then choose whether you need the nth term or the sum.
| Question asks for... | Use... | Why |
|---|---|---|
| Value after n steps | nth term | You want one term |
| Total after many steps | sum formula | You want many terms added |
Worked example:
- A worker saves 120 in month 1, 150 in month 2, 180 in month 3, ...
- This is arithmetic with common difference 30.
- To find the 10th month, use the nth term.
- To find the total saved in 10 months, use the sum formula.
Exam Tips:
- Ask: do I need one term or a total?
- Write down a, d, and n before using a formula.
- Keep the meaning of each term clear.
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
Approximate arithmetic models
Real data is not always perfect: In real life, values may be close to arithmetic but not exact.
If the changes are almost equal, you may still use an arithmetic model as an approximation.
Not exactly arithmetic
- 12, 15, 19, 22
- Differences are 3, 4, 3
- Not perfect
Reasonable model
- The changes stay close to +3
- An arithmetic model may still be useful
- State that it is approximate
When IB likes this
- Tables from real life
- Predictions using a simple model
- Explaining whether a model is reasonable
Finding u₁ and d from two non-first terms
What kind of question is this?
- IB gives you two middle terms — not u₁. For example: salary in year 3 and salary in year 8.
- You need to find what they earned in year 1, or the total over several years.
- You cannot read off u₁ or d directly — you have to solve for them.
How to solve it
- Substitute each term into uₙ = u₁ + (n − 1)d — you get two equations, both containing u₁ and d.
- Label them equation (1) and equation (2).
- Subtract the smaller equation from the larger — u₁ cancels, leaving only d.
- Solve for d.
- Substitute d back into either equation to find u₁.
Find d and u₁ from two middle terms
Priya joins a company in 2015. In her 3rd year she earns 43 200. Find d and u₁.
Step by step
- Write the formula for each year: uₙ = u₁ + (n − 1)d
- Year 8 (n = 8): u₁ + 7d = 43 200 — call this equation (1)
- Year 3 (n = 3): u₁ + 2d = 31 200 — call this equation (2)
- Subtract (2) from (1) to cancel u₁:
- 5d = 12 000 → d = 2 400
- Substitute into equation (2): u₁ + 4 800 = 31 200 → u₁ = 26 400
Final answer
d = 26 400
Which year does salary first exceed $55 000?
Using u₁ = 26 400 and d = 2 400, find the first year Priya's salary exceeds $55 000.
Step by step
- Set uₙ > 55 000:
- 26 400 + (n − 1) × 2 400 > 55 000
- (n − 1) × 2 400 > 28 600 → n − 1 > 11.92...
- n > 12.92... → round up: n = 13
- n = 13 means year 13 from 2015: 2015 + 12 = 2027
Final answer
2027
Always round up for threshold questions: n = 12.92 means the salary reaches the threshold partway through year 12. The question asks when it first exceeds — that is year 13. Always round up, never round normally.
Exam Tips:
- Write both equations and label them (1) and (2) — the method mark is for showing the subtraction step.
- Show the decimal value of n before rounding — that line earns a mark.
- Convert n to a calendar year using the starting year given in the question.
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Quick recap
- Simple interest gives equal increase each time.
- Equal increase means arithmetic model.
- Use nth term for one value.
- Use sum formula for a total.
- Real data may be only approximately arithmetic.
- If two middle terms are given, write two equations and subtract to find d.
1-minute micro-practice:
- A balance grows by 40 each year. Is this arithmetic? Answer: yes.
- A sequence is 500, 560, 620, 680, ... Find the common difference. Answer: 60.
- Do you use nth term or sum to find the 12th value? Answer: nth term.
- Do you use nth term or sum to find the total of the first 12 values? Answer: sum.
Find d from two middle terms
Sofia saves the same extra amount each month.__LINEBREAK_____LINEBREAK__In month 3 she saves 135.__LINEBREAK_____LINEBREAK__Find d.
Step by step
- Write the formula: uₙ = u₁ + (n − 1)d
- Substitute n = 3 and uₙ = 95: u₁ + 2d = 95 — equation (1)
- Substitute n = 7 and uₙ = 135: u₁ + 6d = 135 — equation (2)
- Subtract (1) from (2): 4d = 40
- d = 10 per month
Final answer
d = $10
Exam Tips:
- Translate the story into a sequence first.
- Check whether the increase is constant.
- Say if the model is approximate.
- If given two non-first terms, write uₙ = u₁ + (n−1)d twice and subtract — this cancels u₁ and leaves d.