Step by step
The question
Question: a₁ = −6 and aₙ = aₙ₋₁ + 5.
Write the explicit rule: aₙ with n in it.
aₙ₋₁ means the term just before aₙ.
A rule that makes each term from the term before is called recursive.
1 Read the rule
The first terms: −6, −1, 4.
We add the same number every time.
So it is arithmetic, with d = 5.
2 Write the explicit rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Careful!
Not 5n plus the start.
Term 1 adds no jumps yet, so we add d only n − 1 times.
Put in a₁ = −6 and d = 5.
The bracket is times 5.
n times 5 is 5n.
−1 times 5 is −5.
Plain numbers go together: −6 − 5 is −11.
So aₙ = 5n − 11.
3 Check
Let's check with term 3.
The recursive rule gave 4.
The explicit rule with n = 3 gives 4 too.
It works!
Let's look back.
First: read the rule.
Then: write the explicit rule.
Last: check.