Step by step
The question
Question: aₙ = 5n − 4.
Write the recursive rule: each term from the term before.
an = 5n − 4
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 1.
a1 = 5(1) − 4 = 1
n = 2 gives 6.
a2 = 5(2) − 4 = 6
n = 3 gives 11.
a3 = 5(3) − 4 = 11
2 Find the jump
The list: 1, 6, 11.
= 1, 6, 11, ...
So each term is the term before, plus 5.
6 − 1 = 5
3 Write the recursive rule
The start is a₁ = 1.
The rule: aₙ = aₙ₋₁ + 5.
= a1 = 1, an = an−1 + 5
4 Check
Let's check with term 4.
The explicit rule gives 16.
explicit: a4 = 5(4) − 4 = 16
The recursive rule: the term before is 11, so 11 + 5 is 16.
The same.
It works!
recursive: a4 = 11 + 5 = 16
Let's look back.
First: find the first terms.
Then: find the jump.
Then: write the recursive rule.
Last: check.
Answer
= a1 = 1, an = an−1 + 5