a_1 = 7, a_n = a_(n-1) - 2, write the explicit rule

Step by step

The question

Question: a₁ = 7 and aₙ = aₙ₋₁ − 2.

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: 7, 5, 3.

We add the same number every time.

So it is arithmetic, with d = −2.

2 Write the explicit rule

The rule from the sequences lesson: the first term, plus (n − 1) times the difference.

Careful!

Not −2n plus the start.

Term 1 adds no jumps yet, so we add d only n − 1 times.

Put in a₁ = 7 and d = −2.

The bracket is times −2.

n times −2 is −2n.

−1 times −2 is 2.

Plain numbers go together: 7 + 2 is 9.

So aₙ = −2n + 9.

3 Check

Let's check with term 3.

The recursive rule gave 3.

The explicit rule with n = 3 gives 3 too.

It works!

Let's look back.

First: read the rule.

Then: write the explicit rule.

Last: check.