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Find when a sequence first reaches or passes a given value

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Which method identifies the first term that satisfies a threshold condition?

\[t_0=10,\quad t_{n+1}=t_n+7,\quad \text{threshold}=40\]
Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

Finding the first term past a threshold

A threshold question asks for the smallest whole term number that satisfies a condition. Compare the term just before the crossing with the term at the crossing, then report the number of applications

\[t_n=t_0+nD\quad\text{or}\quad t_n=R^n\times t_0\]
  1. Start at term zero The starting value is term 0, so the index counts applications of the rule \(\displaystyle t_0=10,\quad t_1=17\)
  2. Count applications The term number tells how many times the rule has been applied, not how many values have been listed \(\displaystyle t_5\text{ means 5 applications of the rule}\)
  3. Compare before and at Check the term immediately before a possible crossing and the term at the crossing \(\displaystyle t_4=38,\quad t_5=45,\quad 38\le40<45\)
  4. Use the first whole index Choose the smallest whole term number that satisfies the stated condition \(\displaystyle n>7\quad\Rightarrow\quad n=8\)

Worked example

Find the first term number at which the sequence exceeds 50

\[t_0=8,\quad t_{n+1}=t_n+6,\quad t_n>50\]
  1. \(\displaystyle t_n=t_0+nD=8+6n\)Use the arithmetic closed rule with starting value 8 and common difference 6.
  2. \(\displaystyle 8+6n>50\Rightarrow6n>42\Rightarrow n>7\)Solve the inequality for the term number
  3. \(\displaystyle n=8\)The smallest whole number greater than 7 is 8.
  4. \(\displaystyle t_7=50,\quad t_8=56\)Term 7 is not above 50, while term 8 is above 50, so the crossing is confirmed

Common mistakes

Rounding towards the threshold
✕\(\displaystyle t_4=38\Rightarrow n=4\)✓\(\displaystyle t_4=38,\quad t_5=45\Rightarrow n=5\)
The term before the crossing does not satisfy the condition, so it cannot be the first term past the threshold
Counting listed terms
✕\(\displaystyle t_0,t_1,t_2,t_3,t_4,t_5\text{ gives }n=6\)✓\(\displaystyle t_5\text{ is reached after }5\text{ applications}\)
The starting term is numbered zero, so six listed terms contain only five applications
Starting at term one
✕\(\displaystyle 10=t_1,\quad17=t_2,\quad24=t_3\)✓\(\displaystyle 10=t_0,\quad17=t_1,\quad24=t_2\)
The first value is term 0, so assigning it term 1 shifts every later term number by one

Difficulty levels

introArithmetic growth with a whole-number rule, a small crossing index and repeated applications
standardArithmetic decay or geometric growth with a crossing in a money model measured in years
challengeGeometric decay, sub-annual index units, threshold boundaries and a conversion to years
examA choice of term numbers containing the index before and after the first satisfying term

Questions about this topic

What does first exceed mean?+

It means the smallest term number whose value is greater than the stated threshold, not a term equal to or below it

Why compare two neighbouring terms?+

The term before the crossing must fail the condition and the term at the crossing must satisfy it, which proves that the selected term is first

Why is the starting term t₀?+

The index counts applications of the rule, and no application has occurred at the starting value, so its index is zero

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