Link recurrence, explicit, tabular and graphical forms practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Linking recurrence representations

Equivalent representations describe the same values at the same indices. Identify a fixed difference or ratio, keep V_0 at index zero, and check a later value before selecting a representation

V_0=100,\quad V_{n+1}=1.2V_n,\quad V_n=100(1.2)^n

Keep the initial value at index zero The starting value belongs to V_0, and every equivalent representation must use the same index $\displaystyle V_0=100$
Test a fixed difference or ratio Subtract consecutive terms for an additive model, or divide consecutive non-zero terms for a multiplicative model $\displaystyle 120-100=20,\quad \frac{120}{100}=1.2$
Write the matching recurrence and rule Use a fixed difference d in an additive recurrence and a fixed multiplier R in a multiplicative recurrence $\displaystyle V_{n+1}=V_n+d,\quad V_n=V_0+nd,\quad V_{n+1}=RV_n,\quad V_n=V_0R^n$
Check integer-indexed values A sequence graph shows separate points at whole-number indices, and a later value should agree with the same rule $\displaystyle V_3=100(1.2)^3=172.8$

Worked example

A savings balance is $850 at month zero and increases by $75 each month. Write its recurrence and explicit rule, then find the balance at month four

V_0=850,\quad d=75

$\displaystyle V_{n+1}=V_n+75$The fixed monthly increase gives an additive recurrence
$\displaystyle V_n=850+75n$The explicit rule starts at 850 at month zero and adds 75 for each month
$\displaystyle V_4=850+75(4)=1150$Substitute month four into the explicit rule to find 1,150 dollars

Common mistakes

Using a difference for a constant-ratio sequence

✕$\displaystyle V_{n+1}=V_n+20$✓$\displaystyle V_{n+1}=1.2V_n$

A constant-ratio sequence changes by a percentage of the current value, so subtracting terms does not give its rule

Shifting the initial index

✕$\displaystyle V_n=100(1.2)^{n-1}$✓$\displaystyle V_n=100(1.2)^n$

The initial value is V_0, so n transitions require the multiplier to the power n

Reversing the ratio

✕$\displaystyle R=\frac{100}{120}=\frac{5}{6}$✓$\displaystyle R=\frac{120}{100}=1.2$

A forward multiplier is the next term divided by the preceding term

Difficulty levels

introMatch a term table to an additive recurrence

standardMatch a recurrence to an explicit rule or discrete graph

challengeMatch multiplicative decay representations with close direction and indexing distractors

examSelect an equivalent representation from a regenerated table, recurrence, explicit rule or graph

Questions about this topic

How can an additive sequence be recognised?+

Subtract each term from the next term. A constant difference indicates an additive recurrence

How can a multiplicative sequence be recognised?+

Divide each non-zero term by the preceding term. A constant ratio indicates a multiplicative recurrence

Why are sequence graph points not joined?+

The recurrence defines values at whole-number indices only. Joining the points would suggest values between indices that the sequence has not defined

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