New
New
Year 8

# Finding the nth term

I can find the n^th term rule by investigating the common difference.

New
New
Year 8

# Finding the nth term

I can find the n^th term rule by investigating the common difference.

## Lesson details

### Key learning points

1. Finding the common difference can help when finding the n^th term rule.
2. Comparing the sequence to an appropriate multiplication table can help identify the translation that has been made.
3. The n^th term can be found for all arithmetic sequences.
4. The n^th term rule can be used to identify the term number of a given number in a sequence.

### Common misconception

That the sequence 6,11,16,21, ... is 5n+6 because it goes up by 5 and starts at 6.

Compare 6,11,16,21, ... to 5,10,15,20, ... "What is the shift? The translation? If that is 5n then this is 5n with how much more?"

### Keywords

• N^th term - The nth term of a sequence is the position of a term in a sequence where n stands for the term number.

Get students to explain back to you why 6,11,16,21, ... is 5n+1 and NOT 5n+6. If they can counter-argue the misconception then they truly understand what is going on here.
Teacher tip

### Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

## Starter quiz

### 6 Questions

Q1.
The n$$^\text{th}$$ term is the position of a term in a sequence ($$n$$ is the term number). It can be used to calculate any term so is also known as a __________ for finding the n$$^\text{th}$$ term.
sequence
variable
Q2.
What is the term-to-term rule for this pattern sequence?
Multiply by 3
Goes up by 1
Multiply by 3 and add 1
Q3.
What is the position-to-term rule for this pattern sequence?
Multiply the term number by 3
Add 1 to the previous term
Add 3 to the previous term
Q4.
Which of these are arithmetic (linear) sequences?
1, 2, 4, 8, ...
1, 4, 8, 13, ...
Correct answer: 1, 4, 7, 10, ...
Correct answer: 10, 7, 4, 1, ...
Correct answer: 0.4, 0.7, 1, 1.3, ...
Q5.
This pattern represents people seated around an increasing number of tables. If you were asked to find the number of people around 50 tables, which calculation would you do?
$$50 \times 4$$
$$50 \times 2 + 4$$
$$50 \times 6$$
$$4 + 4 + 4 + 4 + 4 + \;$$... fifty times.
Correct answer: $$50 \times 4 + 2$$
Q6.
Order these arithmetic sequences in terms of the size of their common difference. Start with the greatest common difference.
1 - -101, -87, -73, -59, ...
2 - 124, 133, 142, 151, ...
3 - -8, 0, 8, 16, 24, ...
4 - -5, 2.5, 10, 17.5 ...
5 - 1852, 1859, 1866, 1873, ...
6 - 1896, 1888, 1880, 1872, ...
7 - 2190, 2180, 2170, 2160, ...

## Exit quiz

### 6 Questions

Q1.
5$$n$$ - 2 is the __________ of the sequence 3, 8, 13, 18, ...
expression
Correct answer: $$n^\text{th}$$ term
unknown
term-to-term rule
Q2.
To find the $$n^\text{th}$$ term of an arithmetic sequence, which of the following do we need?
The 1$$^\text{st}$$ term
The last term
The increase.
Q3.
17, 21, 25, 29, ... is a translation of what from the sequence 4$$n$$?
+17
+21
It is not a translation of 4$$n$$.
+4
Q4.
What is the $$n^\text{th}$$ term of the arithmetic sequence 10, 15, 20, 25, ...?
$$5n+10$$
Correct answer: $$5n+5$$
$$10n+5$$
$$10n-5$$
+5
Q5.
The sequence $$17-7n$$ has a common difference of what?
17
-17
7
Find the $$n^\text{th}$$ term of the sequence 2.31, 2.22, 2.13, 2.04, ...
$$0.09n-2.31$$
$$2.31-0.09n$$
$$-0.09n$$
Correct answer: $$2.4-0.09n$$
$$2.04+0.09n$$