Growth and financial problems

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What this lesson covers

  • A fixed added amount is arithmetic.
  • A percentage change is geometric.
  • Regular deposits form a geometric series.
Key idea

Which kind is it

  • A fixed extra amount each time gives an arithmetic sequence.
  • A percentage increase multiplies by 1 + the rate, which is geometric.
Worked example

Saving a bit more each month

  • Save $100 in month 1 and $20 more each month after.
  1. a = 100, d = 20
  2. t₁₂ = 100 + 11 × 20 = $320
  3. S₁₂ = 6(200 + 220)
  4. S₁₂ = $2520
Worked example

Compound interest

  • $5000 at 4% a year for 5 years.
  1. r = 1.04
  2. 5000 × 1.04⁵
  3. = $6083.26
Worked example

Regular deposits

  • $1000 at the start of each year at 5% a year. What is it worth after 3 years?
  1. 1000(1.05 + 1.05² + 1.05³)
  2. a = 1050, r = 1.05, n = 3
  3. S₃ = 1050(1.05³ − 1) ÷ 0.05
  4. S₃ = $3310.13

Common mistakes

  • Using r = 0.04 for 4% growth.
    Growth multiplies by 1.04.
  • Forgetting a bouncing ball travels up and down.
    After the drop, each rebound counts twice.

Your turn

  • 1. A town of 2000 grows 3% a year. Estimate the population after 10 years.
  • 2. A ball drops 2 m and each rebound is 3/4 of the last height. Find the total distance travelled.
Show answer

1. 2000 × 1.03¹⁰, about 2688 2. 2 + 2 × (1.5 ÷ 0.25) = 14 m

Summary

What to take away

  • Added amount: arithmetic. Percentage: geometric.
  • Write the first few terms when unsure.

More in AS 91258

Arithmetic sequences and seriesNotesGeometric sequences and seriesNotesSigma notationNotesGrowth and financial problemsNotes

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