Step by step
The question
Question: aₙ = −2n + 10.
Write the recursive rule: each term from the term before.
an = −2n + 10
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 8.
a1 = −2(1) + 10 = 8
n = 2 gives 6.
a2 = −2(2) + 10 = 6
n = 3 gives 4.
a3 = −2(3) + 10 = 4
2 Find the jump
The list: 8, 6, 4.
= 8, 6, 4, ...
So each term is the term before, minus 2.
6 − 8 = −2
3 Write the recursive rule
The start is a₁ = 8.
The rule: aₙ = aₙ₋₁ − 2.
= a1 = 8, an = an−1 − 2
4 Check
Let's check with term 4.
The explicit rule gives 2.
explicit: a4 = −2(4) + 10 = 2
The recursive rule: the term before is 4, so 4 − 2 is 2.
The same.
It works!
recursive: a4 = 4 − 2 = 2
Let's look back.
First: find the first terms.
Then: find the jump.
Then: write the recursive rule.
Last: check.
Answer
= a1 = 8, an = an−1 − 2