How do annuity investments and savings plans grow when regular payments are added to a compounding balance, and how is the finance solver set up for them?
Model an annuity investment or savings plan where a regular payment is added to a compounding balance using the recurrence relation, analyse it on a finance solver, and find the final balance, total interest and required payment
A focused answer to the VCE General Mathematics Unit 3 Recursion and financial modelling key-knowledge point on annuity investments. Regular additions to a compounding balance, the growth recurrence, finance solver sign conventions, final balance, total interest, and solving for the required payment.
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What this dot point is asking
VCAA wants you to model a savings plan, called an annuity investment, where you add a regular payment to a balance that also earns compound interest. Each period the balance is multiplied by the growth factor and then a deposit is added. You set this up as a recurrence, analyse it on the finance solver, and find the final balance, the total interest earned, or the regular payment needed to reach a target. This is the savings mirror image of a reducing-balance loan.
The growth recurrence
Let be the initial deposit, the growth factor per period (with the nominal annual rate and compounds per year), and the regular payment added each period. Then
This is a mixed recurrence: a geometric multiplication plus a constant addition. It is the same structure as a reducing-balance loan, except here the constant term is added rather than subtracted, because deposits build the balance up.
Using the finance solver
On a CAS finance solver you enter the number of payments , the annual rate, the present value, the payment, the future value, the payments per year and the compounds per year. The sign convention matters: money you put into the investment (the present value deposit and the regular payments) is usually entered as negative, and the future value you receive is positive. Set the unknown field to solve.
Solving for the required payment
A common question gives a savings target and asks for the regular deposit needed. You enter the target as the future value, the known present value and rate, and solve for the payment. The solver returns the deposit per period that reaches the goal.
Why this matters for the exams
Annuity investment questions appear most years and reward correct solver setup and a clean interest calculation. They contrast neatly with reducing-balance loans: same mixed recurrence, opposite direction of the constant term. Show the recurrence or the solver inputs explicitly, and state the total interest as balance minus principal minus deposits so the marker can follow your reasoning.
Exam-style practice questions
Practice questions written in the style of VCAA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2025 VCAA1 marksRita opens a savings account with an initial deposit of 50 into the account. Assume there are exactly 52 weeks in one year. The annual interest rate, compounding weekly, that is required to achieve a balance of $14 000 after three years is closest to A. 8.4% B. 14.2% C. 14.6% D. 17.2%Show worked answer →
This is a savings plan (annuity investment): a compounding balance with a regular $50 added each week. It is set up on a finance solver with N = 3 x 52 = 156 weekly periods, PV = -4000, PMT = -50 (both paid in), FV = 14 000, P/Y = C/Y = 52, then solve for the annual rate.
The solver returns an annual interest rate of about 8.4%.
You can check it: at 8.4% per annum the weekly rate is 0.084 / 52, and compounding 50 added for 156 weeks gives about 14 000. So the answer is A.