Key points about sequences
A number sequence is a list of ordered numbers that follow a pattern or a rule. A term-to-term rule explains how to find the next A value in a sequence. The 3rd term is the 3rd value in the sequence. of a sequence.
The ๐th term of a sequence is a โposition-to-term rule that can be used to find out any term in a sequence.
The ๐th term of an A sequence that increases or decreases by the same number each time, eg 4, 7, 10, 13. Sometimes called a linear sequence.ย (sometimes known asโ aโ โlinearโ sequence) is found by comparing the sequence to an appropriate times table.
Higher tier - The ๐th term of a Describing an expression of the form ๐๐ฅยฒ + ๐๐ฅ + ๐ where ๐, ๐ and ๐ are integers. sequence is found by considering the Once the first difference between values of a sequence is calculated, the second difference is the difference between these values.ย between the terms and comparing the sequence to another that contains ๐ยฒ.
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Term-to-term rules
A sequence is a list of values that follow a rule. Each value is called a term.
A rule that explains how to find the next term in a sequence is called a term-to-term rule.
The most common type of sequence is an arithmetic (or linear) sequence. The difference between each term is the same every time, and is known as the common difference.
Follow the working out below
GCSE exam-style questions
- โ20, โ17, โ14, โ11 are the first four terms of a sequence. Find the term-to-term rule.
The term-to-term rule is โadd 3โ.
The negative numbers are getting closer to zero and the terms are increasing by 3 each time.
- The term-to-term rule of a sequence is โmultiply by 5, then subtract 3โ. The first term is 2. Work out the 3rd term.
The third term is 32
To work out the second term, multiply the first term by 5, then subtract 3.
2 ร 5 โ 3 = 7โ.โ
To work out the third term, multiply the second term by 5, then subtract 3.
7 ร 5 โ 3 = 32โ.โ
- The term-to-term rule of a sequence is โadd 3 then multiply by 2โ. The third term is 46. Calculate the first term.
The first term is 7.
- To calculate the second term, work backwards from the third term. The term-to-term rule in reverse is โdivide by 2, then subtract 3โ.โโ
46 รท 2 = 23. Then 23 โ 3 = 20. The second term is 20.
- To calculate the first term, work backwards from the second term. Use the same term-to-term rule in reverse.
20 รท 2 = 10. Then 10 โ 3 = 7. The first term is 7.
๐th term rules
Rather than finding the next term or next two terms of a sequence, it may be necessary to work out the 50th term, for example.
To do this without writing out all 50 terms, a general rule called the ๐th term is found.
To find an A mathematical sentence expressed either numerically or symbolically made up of one or more terms, eg 8 + 2, or 6๐ฅ, or 5๐ฅยฒ + 3๐ฆ, or 3๐๐๐. for the ๐th term of an A sequence that increases or decreases by the same number each time, eg 4, 7, 10, 13. Sometimes called a linear sequence.ย , work out the The difference between each term in an arithmetic linear sequence. between the terms and treat the sequence as a times table that has been shifted.
- For example, 3, 7, 11, 14 has a common difference of 4, and is the 4 times table with 1 subtracted. The ๐th term is therefore 4๐ โ 1.
To find the 50th term, In algebra, to replace a letter with a number. the value of 50 into the ๐th term rule.
Follow the working out below
GCSE exam-style questions
- A linear sequence starts 3, 8, 13, 18, 23. Work out an expression for the ๐th term.
5๐ โ 2
Write the numbers for ๐ above the sequence.
The common difference between the terms is 5, โso write the 5 times table under โClick here to enter text.โโ the values of ๐โ. Label the 5 times table as 5๐.
Work out how to get from the 5๐ row to the sequence. The sequence is the 5 times table subtract 2. The ๐th term is 5๐ โ 2.
- An arithmetic sequence starts 14, 11, 8, 5, 2. Work out an expression for the ๐th term.
โ3๐ + 17
- Write the numbers for ๐ above the sequence.
- The common difference is โ3, so write the โ3 โtimes tableโ under the values of ๐. Label the row as โ3๐.
- Work out how to get from the โ3๐ row to the sequence. The โ3๐ row has had 17 added.
The ๐th term is โ3๐ + 17.
(Another way to write the answer is 17 โ 3๐).
- โWrite down the first three terms of a sequence where the ๐th term is given by ๐ยฒ + 5.
6, 9, 14
โ1. The first term is when ๐ = 1. Substitute 1 into ๐ยฒ + 5 to give 12 + 5 = 6.
- โThe second term is when ๐ = 2. Substitute 2 into ๐ยฒ + 5 to give 22 + 5 = 9.
โ3. The third term is when ๐ = 3. Substitute 3 into ๐ยฒ + 5 to give 32 + 5 = 14.
โ4. The first three terms are 6, 9, 14.
โ(Note: this sequence is not an arithmetic sequence as it does not go up by the same number each time.)
Quiz โ Sequences
Practise what you have learned about sequences with this quiz.
Higher โ ๐th term of a quadratic sequence
Quadratic sequences have an ๐th term rule that contains ๐ยฒ.
Follow the working out below
Example 1
Example 2
The differences between the terms are not equal, but the Once the first difference between values of a sequence is calculated, the second difference is the difference between these values.ย between the terms are equal.
To find the ๐th term, follow these steps:
Work out the first differences between the terms. The first differences are not the same. Work out the second differences.
The second differences will be the same. The A number or symbol multiplied with a variable or an unknown quantity in an algebraic term. For example, 4 is the coefficient of 4nยฒ. (๐) of ๐ยฒ in the ๐th term rule is always half of the second difference.
Compare the numbers of the sequence ๐๐ยฒ with the original quadratic sequence. The difference between them will be a A number that does not vary. Constants are different to variables such as ๐ฅ and ๐ฆ that can take many values.ย , or should make an A sequence that increases or decreases by the same number each time, eg 4, 7, 10, 13. Sometimes called a linear sequence.ย .
Add the constant or ๐th term for the arithmetic sequence to ๐๐ยฒ to give the ๐th term for the quadratic sequence.
GCSE exam-style questions
- A quadratic sequence has an ๐th term of 2๐ยฒ + 4๐ โ 3. Find the first 3 terms.
3, 13, 27
Substitute ๐ = 1 into the expression. The first term is 2(1ยฒ) + 4(1) โ 3 = 2(1) + 4 โ 3 = 3
Substitute ๐ = 2 into the expression. The second term is 2(2ยฒ) + 4(2) โ 3 = 2(4) + 8 โ 3 = 13
Substitute ๐ = 3 into the expression. The third term is 2(3ยฒ) + 4(3) โ 3 = 2(9) + 12 โ 3 = 27
- Work out the ๐th term of the sequence 6, 9, 14, 21, 30.
๐ยฒ + 5
The first differences are + 3, + 5, + 7, + 9.
The second differences are all +2. Half of 2 is 1, so the coefficient of ๐ยฒ is 1.
Write the values of 1๐ยฒ and compare it to the sequence. The constant value of 5 is always added to ๐ยฒ to make the sequence.
Write the final ๐th term rule as ๐ยฒ + 5.
- Work out the โโ๐โโth term of the sequence 3, 9, 17, 27, 39.โโ
โโ๐โโยฒ + 3โโ๐โโ โโโ 1
The first differences are + 6, + 8, + 10, + 12. โโ
The second differences are all + 2. Half of 2 is 1, so the coefficient of โโ๐โโยฒโโ is 1.โโ
Write the values of โโ1๐โโยฒโโ and compare it to the sequence. The differences 2, 5, 8, 11, 14 form an arithmetic sequence whose ๐th term is โโ3๐ โ1โโ (the 3 times table subtract 1).โโ
Add 3๐ โ 1 to ๐โโยฒโโ to give the final ๐th term rule
โโ๐โโยฒ + 3๐โโ โโโ 1.โโโโ
Quadratic sequences โ interactive activity
This interactive activity will help you to learn how to create quadratic sequences by selecting different coefficient values.
Higher โ Quiz โ Sequences
Practise what you have learned about sequences with this quiz for Higher tier.
Now that you have revised sequences, why not try looking at geometric and special sequences?
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