Step by step
The question
Question: aₙ = 5 × (−2)ⁿ ⁻ ¹.
Write the recursive rule: each term from the term before.
an = 5 × (−2)n − 1
Careful!
A recursive rule needs two parts: the start a₁, and how to get the next term.
Without a₁ we cannot begin.
Careful: a recursive rule needs its start, a₁
1 Find the first terms
We put n = 1, 2 and 3 into the rule.
n = 1 gives 5.
a1 = 5 × (−2)0 = 5
n = 2 gives −10.
a2 = 5 × (−2)1 = −10
n = 3 gives 20.
a3 = 5 × (−2)2 = 20
2 Find the jump
The list: 5, −10, 20.
= 5, −10, 20, ...
So each term is the term before, times −2.
−10 ÷ 5 = −2
3 Write the recursive rule
The start is a₁ = 5.
The rule: aₙ = −2aₙ₋₁.
= a1 = 5, an = −2an−1
4 Check
Let's check with term 4.
The explicit rule gives −40.
explicit: a4 = 5 × (−2)3 = −40
The recursive rule: the term before is 20, so −2 × 20 is −40.
The same.
It works!
recursive: a4 = −2 × 20 = −40
Let's look back.
First: find the first terms.
Then: find the jump.
Then: write the recursive rule.
Last: check.
Answer
= a1 = 5, an = −2an−1