Mathematics Advanced • Year 12 • Module 7 • Lesson 7
Introducing Annuities — Future Value
Practise HSC-style writing on FV annuities, including a structured GP-derivation extended response.
1. Short-answer questions
1.1 Tahlia deposits $250 at the end of each month into a savings account paying 4.8% p.a. compounded monthly. Find the value of her account after 6 years, correct to the nearest cent. 2 marks Band 3
1.2 Connor wants to have $40,000 in 8 years' time to put towards a house deposit. He plans to make equal end-of-month contributions into an account paying 5.4% p.a. compounded monthly. Find the monthly contribution required, correct to the nearest dollar. 3 marks Band 3-4
1.3 $500 is deposited at the end of each year into an account paying 6% p.a.
(a) Find the total contributions and the FV after 10 years.
(b) Hence find the interest earned and the percentage of FV that is interest. 4 marks Band 4
2. Extended response
2.1 An ordinary annuity has end-of-period contributions of $a, an interest rate r per period, and runs for n periods.
(a) Write the FV as the sum of the future values of the n individual contributions, and identify it as a geometric series. State the first term, common ratio, and number of terms.
(b) Using the GP sum formula S_n = a(r^n − 1)/(r − 1), derive the FV annuity formula FV = a × [(1 + r)ⁿ − 1] / r.
(c) Hence find the future value of $600 contributed at the end of each month into a super fund paying 7.2% p.a. compounded monthly over 30 years. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — writes FV = a + a(1+r) + a(1+r)² + … + a(1+r)ⁿ⁻¹ (i.e. recognises that the last contribution earns no interest and the first earns n − 1 periods).
• 1 mark — identifies first term a, common ratio (1 + r), n terms.
Part (b) — 3 marks
• 1 mark — substitutes a = a, ratio = (1+r), n = n into S_n correctly.
• 1 mark — simplifies (1+r) − 1 = r in the denominator.
• 1 mark — writes the final boxed formula FV = a × [(1+r)ⁿ − 1] / r.
Part (c) — 2 marks
• 1 mark — converts to monthly: r = 0.072/12 = 0.006, n = 360.
• 1 mark — correct numerical FV.
Your response:
Stuck on (a)? The contribution at the end of period k earns interest for the remaining (n − k) periods.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Monthly savings (2 marks)
Sample response. r = 0.048 / 12 = 0.004; n = 6 × 12 = 72. FV = 250 × [(1.004)⁷² − 1] / 0.004 = 250 × (1.33399 − 1)/0.004 = 250 × 83.4972 = $20,874.31.
Marking notes. 1 mark — correct conversion of r and n to monthly. 1 mark — correct substitution and numerical answer to the nearest cent. Answers within $0.10 due to rounding accepted.
1.2 — Required monthly contribution (3 marks)
Sample response. r = 0.054/12 = 0.0045; n = 96. Factor = [(1.0045)⁹⁶ − 1]/0.0045 = (1.53938 − 1)/0.0045 = 119.862. a = 40,000 / 119.862 ≈ $334 per month.
Marking notes. 1 mark — correct r and n. 1 mark — correctly evaluates the factor. 1 mark — transposes and gives a to the nearest dollar. Common error: forgetting to multiply 8 × 12.
1.3 — Contribute, accumulate, decompose (4 marks)
(a) Sample response. Total contributions = 10 × 500 = $5,000. FV = 500 × [(1.06)¹⁰ − 1]/0.06 = 500 × 13.181 = $6,590.40.
(b) Sample response. Interest = 6,590.40 − 5,000 = $1,590.40. Percentage = 1,590.40 / 6,590.40 × 100 ≈ 24.1% of FV is interest.
Marking notes. (a) 1 mark — contributions; 1 mark — correct FV. (b) 1 mark — correct interest; 1 mark — correct percentage to 1 dp.
2.1 — Extended response (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
Part (a). Let a be paid at the end of each period for n periods at rate r per period. The last contribution earns no interest (it is paid at time n), the second-last earns one period, and the very first contribution earns n − 1 periods of interest. Therefore
FV = a + a(1 + r) + a(1 + r)² + … + a(1 + r)ⁿ⁻¹. [1 mark — sum written explicitly.]
This is a geometric series with first term a, common ratio (1 + r), and n terms. [1 mark — GP identified.]
Part (b). Using S_n = a · (R^n − 1) / (R − 1) with first term a, common ratio R = 1 + r, and n terms:
FV = a · [(1 + r)ⁿ − 1] / ((1 + r) − 1). [1 mark — substitution.]
The denominator simplifies because (1 + r) − 1 = r:
FV = a · [(1 + r)ⁿ − 1] / r. [1 mark — denominator simplified.]
This is the FV annuity formula, as required. [1 mark — final boxed result.]
Part (c). Convert to monthly: r = 0.072 / 12 = 0.006, n = 30 × 12 = 360. [1 mark — conversion.]
FV = 600 × [(1.006)³⁶⁰ − 1] / 0.006 = 600 × (8.6133 − 1) / 0.006 = 600 × 1,268.88 ≈ $761,326. [1 mark — numerical answer.]
Total: 7/7.
Band descriptors for marker.
Band 3: Writes the sum but loses an exponent or term; identifies GP partially; attempts (c) without converting to monthly. ≈ 2-3 marks.
Band 4: Sum correct, GP identified; uses GP sum but does not simplify (1 + r) − 1 to r; (c) attempted with correct conversion. ≈ 4-5 marks.
Band 5: Derivation complete and clean; (c) numerically correct but not justified by the derivation. ≈ 5-6 marks.
Band 6: All three parts complete; clearly states first term, ratio, and number of terms; explicit conversion in (c); rounding and units consistent throughout. 7/7.