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General Mathematics/Sequences and finance

Find the rule for a sequence and give the next term

Start a set of 8

1 OF 20

For 3, 6, 12, 24, which test fits every displayed transition?

The table starts at 8. Compare multipliers 1.5, 0.5, 1 and 2. Watch the first two differences as well as the ratio

Scroll sideways to see the whole table.

First difference: 4
Second difference: 6
Ratio of consecutive terms: 1.5

Multiplier1.5

For multiplier 1.5, what do you observe?

Test the same operation across all pairs

For these questions, use a constant add/subtract or multiply/divide rule that fits every displayed term. A finite list alone does not determine a unique rule outside that stated class

1Difference Subtract earlier from later. Equal differences support an additive rule
2Ratio Divide later by earlier, provided the earlier value is not zero. Equal ratios support a multiplicative rule
3Next term Apply the chosen operation to the latest displayed term, not the initial term

Displayed terms or rule: \(\displaystyle 8,4,0,-4\)

Find the repeated additive change for 8, 4, 0, −4.

Answer

Displayed terms or rule: \(\displaystyle 8,4,0,-4\)

Continue 8, 4, 0, −4 using the same additive rule. What comes next?

Answer

Which multiplier is equivalent to 1.5?

Displayed terms or rule: \(\displaystyle 8,12,18,27\)

Continue 8, 12, 18, 27 using the same multiplicative rule

Answer

Displayed terms or rule: \(\displaystyle 24,12,6,3\)

Continue 24, 12, 6, 3 using the same multiplicative rule

Answer

Which two instructions describe the same operation?

Displayed terms or rule: \(\displaystyle 9,6,3,0\)

Use the same additive rule after 9, 6, 3, 0. Give the next term

Answer

Displayed terms or rule: \(\displaystyle 9,6,3,0,-3\)

The sequence is now 9, 6, 3, 0, −3. Give the following term

Answer

A sequence shows 4, 12, 36, 108 and uses a constant multiplier. How do you find its next term?

Without any restriction on the kind of rule, do the first four values prove a unique fifth value?

Check your proposed continuation

Write the repeated operation in words. Check it against every displayed pair, then apply it once for each extra requested term

1Keep the order For ratios use later divided by earlier. For differences use later minus earlier
2Do not divide by zero If an earlier value is zero, do not form that ratio. Test the stated additive rule or another allowed representation
3Interpret cautiously A numerical pattern is not proof that real-world measurements will keep following it

Here is a sequence of numbers: 35.8, 42.8, 49.8, 56.8

Use one constant add/subtract or multiply/divide rule that fits every displayed term

(a)1 mark

State the rule that generates the sequence

Divide by 7 each time

Multiply by 7 each time

Add 7 each time

Subtract 7 each time

(b)1 mark

What is the next term?

Next term

Here is a sequence of numbers: 2,187, 729, 243, 81, 27

Use one constant add/subtract or multiply/divide rule that fits every displayed term

(a)1 mark

What is the rule?

Add 3 each time

Multiply by 3 each time

Subtract 3 each time

Divide by 3 each time

(b)2 marks

What are the next two terms?

Next term Term after that

An amount was saved into an account each period. The amounts were: 224, 336, 504, 756, 1,134

Use one constant add/subtract or multiply/divide rule that fits every displayed term

(a)1 mark

State the rule that generates the sequence

Multiply by 3/2 each time

Add 2 each time

Subtract 2 each time

Divide by 3 each time

(b)1 mark

What is the next term?

Next term

Here is a sequence of numbers: 6,488.1, 2,162.7, 720.9, 240.3, 80.1

Use one constant add/subtract or multiply/divide rule that fits every displayed term

(a)1 mark

Find the rule for the sequence

Add 3 each time

Multiply by 1/3 each time

Divide by 2 each time

Subtract 3 each time

(b)2 marks

What are the next two terms?

Next term Term after that

Here is a sequence of numbers: 66, 62, 58, 54, 50

Use one constant add/subtract or multiply/divide rule that fits every displayed term

1 mark

What term comes next?

Next term

Here is a sequence of numbers: 7, 14, 28, 56

Use one constant add/subtract or multiply/divide rule that fits every displayed term

1 mark

Find the next term in the sequence

63

14

112

28

0/18

Start a set of 8

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

Find and extend a sequence rule

Test the stated class of rules rather than assuming the first difference always works

Additive check Calculate the changes across consecutive pairs \(\displaystyle 9-5=4,\quad13-9=4\)
Multiplicative check Divide each later value by the one before it \(\displaystyle 12/8=18/12=3/2\)
Extend from the latest value Use each new result as the input to the next application \(\displaystyle 27\times(3/2)=40.5\)

Worked example

Use a constant additive or multiplicative rule to extend 6, 9, 13.5, 20.25

6,9,13.5,20.25

\(\displaystyle 9-6=3\)The first additive change is 3.
\(\displaystyle 13.5-9=4.5\)The changes differ, so test ratios
\(\displaystyle 9/6=1.5\)Test the first ratio
\(\displaystyle 13.5/9=1.5\)The next ratio is the same
\(\displaystyle 20.25/13.5=1.5\)All three transitions multiply by 1.5
\(\displaystyle 20.25\times1.5=30.375\)Use the latest displayed term

Common mistakes

Ignoring the numerator

✕\(\displaystyle 3/2=1/2\)✓\(\displaystyle 3/2=1.5\)

Three halves means 3 divided by 2, not one half

Restarting instead of continuing

✕\(\displaystyle 4,12,36,108\Rightarrow12\)✓\(\displaystyle 108\times3=324\)

The next term follows the latest value, not the initial one

Difficulty levels

introChoose an additive rule and extend it once

standardUse additive, multiplicative and divisive rules, including fractional growth

challengeExtend by two terms, handle fractional multipliers and retain signs

examChoose the next term and check all proposed values

Questions about this topic

Can multiplication make positive values smaller?+

Yes. A multiplier between zero and one decreases a positive value. For example, multiplying by one half halves it

Is multiplying by three halves the same as dividing by 2?+

No. Multiply by 3 and divide by 2, or multiply by 1.5. Dividing by 2 alone is multiplication by one half

Why check more than one pair?+

One transition may fit several operations. A proposed rule must fit every displayed transition

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 sequence Find the rule for a sequence and give the next term 3A Sequence patterns and rules: Generate the terms of a sequence from a starting value and a rule 3B Terms and recurrence relations: Generate the terms of a sequence from a starting value and a rule Name and number the terms of a sequence 3B Terms and recurrence relations: Write a recurrence relation for a sequence Find the common difference and the next term of an arithmetic sequence Decide whether a sequence is arithmetic, geometric or neither Tabulate a sequence and graph its terms 3D Arithmetic recurrence and terms: Write a recurrence relation for a sequence Find term n of an arithmetic sequence using the rule 3D Arithmetic recurrence and terms: Find when a sequence first reaches or passes a given value Model a simple-interest investment or loan with a recurrence relation Model flat-rate depreciation with a recurrence relation Model unit-cost depreciation with a recurrence relation Use the rule for term n of a linear growth or decay model Read the financial parameters out of a linear recurrence relation 3E Linear growth and decay in finance: Find when a sequence first reaches or passes a given value Build and compare linear financial models Find the common ratio and the next term of a geometric sequence Tabulate and interpret discrete geometric graphs Distinguish arithmetic and geometric patterns Write and interpret geometric recurrence rules Write an explicit geometric sequence rule Find term n of a geometric sequence using the rule Find the first geometric threshold index 3H Compound interest and depreciation: Find when a sequence first reaches or passes a given value Model a compound-interest investment or loan with a recurrence relation Write a recurrence relation for a loan or investment with a given compounding period Construct complete compound-interest recurrence models Find the first financial threshold period Construct and use reducing-balance depreciation models 3I Geometric models and finance: Find when a sequence first reaches or passes a given value Use the rule for term n of a compound-interest or reducing-balance model Find the interest on a credit-card debt compounding daily Compare the total cost of ways of paying, or the return on ways of investing Find the effect of inflation on a price Find the purchasing power of an amount after inflation

Chapter progress

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