Step by step
The question
Question: a₁ = 3 and aₙ = 2aₙ₋₁.
Write the explicit rule: aₙ with n in it.
a1 = 3, an = 2an−1
aₙ₋₁ means the term just before aₙ.
A rule that makes each term from the term before is called recursive.
an−1 = the term before an
1 Read the rule
The first terms: 3, 6, 12.
= 3, 6, 12, ...
We multiply by the same number every time.
So it is geometric, with r = 2.
× 2 every time: geometric, r = 2
2 Write the explicit rule
The rule from the sequences lesson: the first term, times the ratio to the power (n − 1).
an = a1 × rn − 1
Careful!
The power is n − 1, not n.
Term 1 has no jumps yet: 2 to the power 0 is 1.
Careful: not an = 3 × 2n ✗
Put in a₁ = 3 and r = 2.
So aₙ = 3 × 2ⁿ⁻¹.
an = 3 × 2n − 1
3 Check
Let's check with term 3.
The recursive rule gave 12.
recursive: a3 = 12
The explicit rule with n = 3 gives 12 too.
It works!
explicit: a3 = 3 × 22 = 3 × 4 = 12
Let's look back.
First: read the rule.
Then: write the explicit rule.
Last: check.
Answer
an = 3 × 2n − 1