Solve an inverse geometric finance model practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Q: Why does the model use the period count as the exponent rather than one fewer?

The initial value comes before any change. Each increase in the period count applies the multiplier once more

Q: How is an unknown initial value found?

Raise the known multiplier to the number of periods, then divide the future value by that power. This reverses the multiplication in the geometric model

Q: Why are adjacent periods checked for a threshold?

A period count is a whole number. Showing that the previous period fails and the next period satisfies the condition proves that the next period is the first qualifying one

Q: Can I use a calculator?

A calculator is allowed unless a question is marked no calculator

Q: Which year level and chapter is this?

Year 12, VCE Units 3 & 4, General Mathematics, chapter 7: Recursion and financial models

Solving a geometric finance model

A geometric finance model applies the same multiplier at every period. The zero-indexed rule makes the number of transitions explicit when an initial value, rate or first qualifying period is unknown

V_n=V_0R^n

Start at index zero The initial value is V_0, before any transition has occurred $\displaystyle V_0=P,\quad V_2=PR^2$
Use powers for repeated change A geometric model applies the same multiplier once at every period $\displaystyle V_3=V_0R^3$
Isolate the unknown Rearrange the rule according to whether the initial value, rate or multiplier is unknown $\displaystyle V_0=\frac{V_n}{R^n},\quad R^n=\frac{V_n}{V_0}$
Check a whole-period threshold For an increasing model, compare adjacent whole periods to find the first value above the threshold $\displaystyle V_{n-1}\le T<V_n$

Worked example

An asset is worth $12,000 now and loses 10% of its value each year. Find its value after 3 years

V_0=\$12{,}000,\quad R=1-\frac{10}{100}=0.90

$\displaystyle R=1-\frac{10}{100}=0.90$A 10% annual decrease leaves 90% of the previous value
$\displaystyle V_1=\$12{,}000\left(0.90\right)=\$10{,}800$Apply the reducing multiplier once to find the value after one year
$\displaystyle V_2=\$10{,}800\left(0.90\right)=\$9{,}720$Use the previous value as the input for the second annual transition
$\displaystyle V_3=\$9{,}720\left(0.90\right)=\$8{,}748$Apply the multiplier for the third time to obtain the value after 3 years

Common mistakes

Using n - 1 transitions

✕$\displaystyle V_2=\$1{,}000\left(1.05\right)=\$1{,}050$✓$\displaystyle V_2=\$1{,}000\left(1.05\right)^2=\$1{,}102.50$

V_0 is before any transition, so reaching V_n requires n transitions rather than n - 1.

Accepting a negative financial multiplier

✕$\displaystyle R=-1.05$✓$\displaystyle R=1.05$

A financial multiplier must be positive; growth uses R greater than 1 and decay uses a value between 0 and 1.

Rounding a threshold crossing to the nearest period

✕$\displaystyle n_{\mathrm{cross}}\approx4.25\Rightarrow n=4$✓$\displaystyle V_4=99<100<V_5=103\Rightarrow n=5$

The first qualifying whole period is proved by checking adjacent whole periods, not by rounding a decimal crossing time

Difficulty levels

introFind an unknown initial value with an exact terminating result

standardFind the first whole threshold period by exact iteration

challengeFind an unknown annual rate to a stated decimal place

examChoose and solve the unknown, then interpret a whole-period or context-valid root condition

Questions about this topic

Why does the model use the period count as the exponent rather than one fewer?+

The initial value comes before any change. Each increase in the period count applies the multiplier once more

How is an unknown initial value found?+

Raise the known multiplier to the number of periods, then divide the future value by that power. This reverses the multiplication in the geometric model

Why are adjacent periods checked for a threshold?+

A period count is a whole number. Showing that the previous period fails and the next period satisfies the condition proves that the next period is the first qualifying one

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 7: Recursion and financial models

In this chapter

Iterate a first-order recurrence Link recurrence, explicit, tabular and graphical forms Model and analyse fixed-change finance Compare financial model assumptions and outcomes Model and analyse percentage-change finance Solve an inverse geometric finance model Align nominal rates, period rates and compounding counts Calculate and compare effective annual rates