Geometric tables and discrete graphs
Connect the multiplier, index and term value without confusing a sequence with a continuous curve
- Read the starting index The first term may be labelled 0 or 1. Use the index specified in the question. This topic’s generated questions start at term number 0.
- Multiply successively Use each newly calculated term as the starting point for the next multiplication
- Plot ordered pairs The horizontal coordinate is the term number; the vertical coordinate is its value
- Keep sequence points separate Whole-number term positions do not supply values at fractional positions. Do not add joining segments unless the task explicitly asks for a separate model
- Include the constant boundary For positive terms, a multiplier above 1 gives growth, one between 0 and 1 gives decay, and 1 gives constant values. Constant geometric points lie on a horizontal line, but remain separate points
Worked example
Start at n = 0 with value 16 and multiply by one half. Find the next three terms and represent them as points
\[16,\ 8,\ 4,\ 2\]
- \(\displaystyle 16\times\frac12=8\)This is the value at n = 1.
- \(\displaystyle 8\times\frac12=4\)This is the value at n = 2.
- \(\displaystyle 4\times\frac12=2\)This is the value at n = 3.
- \(\displaystyle (0,16),\ (1,8),\ (2,4),\ (3,2)\)Plot these four separate ordered pairs. No term at a fractional index is supplied
Common mistakes
Index versus position in the list
✕\(\displaystyle (1,12)\)✓\(\displaystyle (0,12)\)
If the first value 12 is explicitly labelled n = 0, its horizontal coordinate is 0.The constant boundary
✕Every geometric sequence must curve upward or downward✓Ratio 1 gives horizontally aligned, separate points
A nonzero constant sequence is both arithmetic and geometricCompare values, not axis coordinates
✕\(\displaystyle R=27\div1\)✓\(\displaystyle R=27\div18\)
The ratio compares consecutive term values. The horizontal coordinate only identifies which term it isDifficulty levels
introComplete tables by repeated multiplication, including decay and ratio 1.
standardMatch a supplied table to an unjoined graph with the correct indices and values
challengeInfer a multiplier from precise supplied graph coordinates
examRecognise a specific repair to a supplied graph. This does not assess independent drawing or formal recurrence construction
Questions about this topic
Does the first term always have index zero?+
No. Read the convention stated in the question. This topic’s generated questions explicitly start at n = 0.
Can a geometric graph look straight?+
Yes. For a nonzero constant sequence the ratio is 1, so the separate points are horizontally aligned
Do graph choices assess drawing?+
They check interpretation and matching. Independent construction on paper or graphing technology needs a separate task
Why do some graphs include an exact coordinate key?+
It supplies precise values when the task asks for a calculation from the graph. The matching choices omit it so the task is not answered by a printed coordinate list
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, General Mathematics, chapter 3: Sequences and finance
In this chapter
Identify the behaviour of a sequenceFind the rule for a sequence and give the next term3A Sequence patterns and rules: Generate the terms of a sequence from a starting value and a rule3B Terms and recurrence relations: Generate the terms of a sequence from a starting value and a ruleName and number the terms of a sequence3B Terms and recurrence relations: Write a recurrence relation for a sequenceFind the common difference and the next term of an arithmetic sequenceDecide whether a sequence is arithmetic, geometric or neitherTabulate a sequence and graph its terms3D Arithmetic recurrence and terms: Write a recurrence relation for a sequenceFind term n of an arithmetic sequence using the rule3D Arithmetic recurrence and terms: Find when a sequence first reaches or passes a given valueModel a simple-interest investment or loan with a recurrence relationModel flat-rate depreciation with a recurrence relationModel unit-cost depreciation with a recurrence relationUse the rule for term n of a linear growth or decay modelRead the financial parameters out of a linear recurrence relation3E Linear growth and decay in finance: Find when a sequence first reaches or passes a given valueBuild and compare linear financial modelsFind the common ratio and the next term of a geometric sequenceTabulate and interpret discrete geometric graphsDistinguish arithmetic and geometric patternsWrite and interpret geometric recurrence rulesWrite an explicit geometric sequence ruleFind term n of a geometric sequence using the ruleFind the first geometric threshold index3H Compound interest and depreciation: Find when a sequence first reaches or passes a given valueModel a compound-interest investment or loan with a recurrence relationWrite a recurrence relation for a loan or investment with a given compounding periodConstruct complete compound-interest recurrence modelsFind the first financial threshold periodConstruct and use reducing-balance depreciation models3I Geometric models and finance: Find when a sequence first reaches or passes a given valueUse the rule for term n of a compound-interest or reducing-balance modelFind the interest on a credit-card debt compounding dailyCompare the total cost of ways of paying, or the return on ways of investingFind the effect of inflation on a priceFind the purchasing power of an amount after inflation