Step by step
The question
Question: a₁ = 4 and aₙ = −2aₙ₋₁.
Find term 6, a₆.
a1 = 4, an = −2an−1
aₙ₋₁ means the term just before aₙ.
A rule that makes each term from the term before is called recursive.
an−1 = the term before an
Careful!
aₙ₋₁ is the term before, not the number n − 1.
For term 2 we put in a₁, which is 4.
Careful: not a2 = −2 × 1 = −2 ✗
1 Start with a₁
The start is given: a₁ = 4.
a1 = 4
2 Use the rule again and again
Term 2: n = 2, so the term before is a₁.
We put in 4.
−2 × 4 is −8.
a2 = −2 × a1 = −2 × 4 = −8
Term 3: the term before is a₂, which is −8.
−2 × (−8) is 16.
a3 = −2 × a2 = −2 × (−8) = 16
Term 4: −2 × 16 is −32.
a4 = −2 × a3 = −2 × 16 = −32
Term 5: −2 × (−32) is 64.
a5 = −2 × a4 = −2 × (−32) = 64
Term 6: −2 × 64 is −128.
a6 = −2 × a5 = −2 × 64 = −128
So a₆ = −128.
3 Check
Let's check by going back.
The rule did times −2.
Going back, we do the opposite: divide by −2.
back: ÷ (−2)
From −128: −128 divided by (−2) is 64, term 5.
−128 ÷ (−2) = 64
64 ÷ (−2) = −32
−32 ÷ (−2) = 16
16 ÷ (−2) = −8
We land on 4, the start.
It works!
−8 ÷ (−2) = 4
Let's look back.
First: start with a₁.
Then: use the rule again and again.
Last: check.
Answer
a6 = −2 × a5 = −2 × 64 = −128