Step by step
The question
Question: aₙ = 3n + 2.
Write the recursive rule: each term from the term before.
an = 3n + 2
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 = 3(1) + 2 = 5
n = 2 gives 8.
a2 = 3(2) + 2 = 8
n = 3 gives 11.
a3 = 3(3) + 2 = 11
2 Find the jump
The list: 5, 8, 11.
= 5, 8, 11, ...
So each term is the term before, plus 3.
8 − 5 = 3
3 Write the recursive rule
The start is a₁ = 5.
The rule: aₙ = aₙ₋₁ + 3.
= a1 = 5, an = an−1 + 3
4 Check
Let's check with term 4.
The explicit rule gives 14.
explicit: a4 = 3(4) + 2 = 14
The recursive rule: the term before is 11, so 11 + 3 is 14.
The same.
It works!
recursive: a4 = 11 + 3 = 14
Let's look back.
First: find the first terms.
Then: find the jump.
Then: write the recursive rule.
Last: check.
Answer
= a1 = 5, an = an−1 + 3