Step by step
The question
Question: aₙ = 2 × 3ⁿ ⁻ ¹.
Write the recursive rule: each term from the term before.
an = 2 × 3n − 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 2.
a1 = 2 × 30 = 2
n = 2 gives 6.
a2 = 2 × 31 = 6
n = 3 gives 18.
a3 = 2 × 32 = 18
2 Find the jump
The list: 2, 6, 18.
= 2, 6, 18, ...
So each term is the term before, times 3.
6 ÷ 2 = 3
3 Write the recursive rule
The start is a₁ = 2.
The rule: aₙ = 3aₙ₋₁.
= a1 = 2, an = 3an−1
4 Check
Let's check with term 4.
The explicit rule gives 54.
explicit: a4 = 2 × 33 = 54
The recursive rule: the term before is 18, so 3 × 18 is 54.
The same.
It works!
recursive: a4 = 3 × 18 = 54
Let's look back.
First: find the first terms.
Then: find the jump.
Then: write the recursive rule.
Last: check.
Answer
= a1 = 2, an = 3an−1