Geometric term rules
Use the stated recurrence and preserve its index origin
- Zero origin For nonzero r, u₀ = a gives uₙ = arⁿ
- Signed multipliers Negative r alternates signs when a is nonzero. Its magnitude controls shrinking or growth
- Zero exceptions Zero start gives all zero terms. Zero ratio gives a followed by zeroes; it cannot be recovered by dividing a zero term
Worked example
A geometric sequence starts at 24 and has a ratio of negative one half. Find the term at index four
\[u_4=24\left(-\frac12\right)^4\]
- \(\displaystyle \left(-\frac12\right)^4=\frac1{16}\)An even number of negative factors gives a positive value
- \(\displaystyle u_4=24\times\frac1{16}=\frac32\)Only r is raised to the fourth power
Common mistakes
Raising the whole product
✕\(\displaystyle u_n=(ar)^n\)✓\(\displaystyle u_n=ar^n\)
The starting value is a single factor; repeated updates multiply by r.Difficulty levels
introCalculate a positive-growth term
standardUse supplied recurrences or listed terms for positive growth and decay
challengeRecover starting values, evaluate multiple terms or recognise positive-ratio behaviour
examSelect a numerical term from supplied choices. Signed and zero cases are also taught here and in sums/limits practice
Questions about this topic
Does the existing numerical term bank include every real ratio?+
No. It provides positive-growth and positive-decay term practice. Signed and zero cases are explicit in this teaching and the sums/limits topic
Can an alternating sequence converge?+
Yes. For example, nonzero starting values multiplied repeatedly by −1/2 tend to zero while alternating signs
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 4: Discrete mathematics
In this chapter
Find term n of an arithmetic sequence using the ruleFind term n of a geometric sequence using the ruleFinite sums and geometric limitsIterate a first-order recurrenceExplicit recurrence solutions, equilibria and population modelsModel and interpret a periodic cash-flow recurrenceEvaluate factorial and combination expressionsCount arrangements with product, sum and restriction rulesCount restricted selections and calculate exact probabilitiesPigeonhole bounds, overlap counts and repeated selectionsRead, transpose and classify a matrixCombine scalar multiples of matricesMultiply matricesDetermine whether an inverse exists and calculate itSolve a matrix equationSolve simultaneous equations using the inverse matrixUse matrix multiplication to answer an application question