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

- Finding the common difference can help when finding the n^th term rule.
- Comparing the sequence to an appropriate multiplication table can help identify the translation that has been made.
- The n^th term can be found for all arithmetic sequences.
- 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).

## Video

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## Starter quiz

Download 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.

Correct answer: formula

formula

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

Correct answer: Add 3

Add 3

Q3.

What is the position-to-term rule for this pattern sequence?

Multiply the term number by 3

Add 1 to the previous term

Correct answer: Multiply the term number by 3 and add 1

Multiply the term number by 3 and add 1

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, ...

1, 4, 7, 10, ...

Correct answer: 10, 7, 4, 1, ...

10, 7, 4, 1, ...

Correct answer: 0.4, 0.7, 1, 1.3, ...

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: $$50 \times 4 + 2$$

$$50 \times 4 + 2$$

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, ...

1

- -101, -87, -73, -59, ...

2 - 124, 133, 142, 151, ...

2

- 124, 133, 142, 151, ...

3 - -8, 0, 8, 16, 24, ...

3

- -8, 0, 8, 16, 24, ...

4 - -5, 2.5, 10, 17.5 ...

4

- -5, 2.5, 10, 17.5 ...

5 - 1852, 1859, 1866, 1873, ...

5

- 1852, 1859, 1866, 1873, ...

6 - 1896, 1888, 1880, 1872, ...

6

- 1896, 1888, 1880, 1872, ...

7 - 2190, 2180, 2170, 2160, ...

7

- 2190, 2180, 2170, 2160, ...

## Exit quiz

Download exit quiz

### 6 Questions

Q1.

5$$n$$ - 2 is the __________ of the sequence 3, 8, 13, 18, ...

expression

Correct answer: $$n^\text{th}$$ term

$$n^\text{th}$$ 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

Correct answer: The common difference

The common difference

The last term

Correct answer: The translation

The translation

The increase.

Q3.

17, 21, 25, 29, ... is a translation of what from the sequence 4$$n$$?

+17

+21

Correct answer: +13

+13

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: $$5n+5$$

$$5n+5$$

$$10n+5$$

$$10n-5$$

+5

Q5.

The sequence $$17-7n$$ has a common difference of what?

17

-17

7

Correct answer: -7

-7

Q6.

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: $$2.4-0.09n$$

$$2.4-0.09n$$

$$2.04+0.09n$$