Worked examples
The question
Question: 2, 5, 8, 11, …: find term 20.
A sequence is a list of numbers in order.
2, 5, 8, 11, ...
1 Find the difference
We add 3 every time.
A list like this is an arithmetic sequence.
The number we add is called the difference: d = 3.
d = 3
2 Find the rule
Term 1 is 2.
a1 = 2
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
an = term number n
Term 2: we add 3 one time.
a2 = 2 + 1 × 3
Term 3: we add it two times.
a3 = 2 + 2 × 3
We add 3 one time less than the term number.
times we add 3 = term number − 1
So for term n, we add 3, (n − 1) times.
an = 2 + (n − 1) × 3
This rule works for every arithmetic sequence: the first term, plus (n − 1) times the difference.
an = a1 + (n − 1) × d
3 Find term 20
For term 20, n is 20.
We put 20 in place of n.
a20 = 2 + (20 − 1) × 3
Brackets come first.
Order: ( ) → x² √ → × ÷ → + −
20 minus 1 is 19.
= 2 + 19 × 3
Times comes before plus.
19 times 3 is 57.
= 2 + 57
2 plus 57 is 59.
a20 = 59
So term 20 is 59.
4 Check
Let's check the rule with a term we know.
Put n = 4.
a4 = 2 + (4 − 1) × 3 = 2 + 3 × 3 = 2 + 9
= 11
We get 11.
That is term 4 in the list.
So the rule is right!
Let's look back.
First: find the difference.
Then: find the rule.
Then: find term 20.
Last: check.
▶ Watch it solved
The question
Question: 5, −10, 20, −40, …: find the rule for term n.
A sequence is a list of numbers in order.
5, −10, 20, −40, ...
1 Find the ratio
We multiply by −2 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = −2.
The minus makes the sign change every time.
r = −2
2 Find the rule
Term 1 is 5.
a1 = 5
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
an = term number n
Term 2: we multiply by −2 one time.
a2 = 5 × (−2)
Term 3: we multiply by −2 two times.
That is (−2)².
a3 = 5 × (−2) × (−2) = 5 × (−2)2
We multiply one time less than the term number.
times we multiply by −2 = term number − 1
So for term n, we multiply by −2, (n − 1) times.
That is the power n − 1.
an = 5 × (−2)n−1
This rule works for every geometric sequence: the first term, times the ratio to the power (n − 1).
So the rule is aₙ = 5 × (−2)ⁿ⁻¹.
an = a1 × rn−1
3 Check
Let's check the rule with a term we know.
Put n = 4.
a4 = 5 × (−2)4−1 = 5 × (−2)3 = 5 × (−8)
= −40
We get −40.
That is term 4 in the list.
So the rule is right!
Let's look back.
First: find the ratio.
Then: find the rule.
Last: check.
▶ Watch it solved
The question
Question: 2, −6, 18, …: find the rule for term n.
A sequence is a list of numbers in order.
2, −6, 18, ...
1 Find the ratio
We multiply by −3 every time.
A list like this is a geometric sequence.
The number we multiply by is called the ratio: r = −3.
The minus makes the sign change every time.
r = −3
2 Find the rule
Term 1 is 2.
a1 = 2
a₁ is term 1, a₂ is term 2.
So aₙ is term number n.
an = term number n
Term 2: we multiply by −3 one time.
a2 = 2 × (−3)
Term 3: we multiply by −3 two times.
That is (−3)².
a3 = 2 × (−3) × (−3) = 2 × (−3)2
We multiply one time less than the term number.
times we multiply by −3 = term number − 1
So for term n, we multiply by −3, (n − 1) times.
That is the power n − 1.
an = 2 × (−3)n−1
This rule works for every geometric sequence: the first term, times the ratio to the power (n − 1).
So the rule is aₙ = 2 × (−3)ⁿ⁻¹.
an = a1 × rn−1
3 Check
Let's check the rule with a term we know.
Put n = 3.
a3 = 2 × (−3)3−1 = 2 × (−3)2 = 2 × 9 = 18
We get 18.
That is term 3 in the list.
So the rule is right!
Let's look back.
First: find the ratio.
Then: find the rule.
Last: check.
▶ Watch it solved
