Given the recursive rule {a_n} = (a_{n – 1}) + 5 and a₁ = 8, find a₅.

The thing about recursive rules is that we need to apply them a bunch of times, over and over, to find the term we want. Get your second term first by substituting 2 for n.

a₂ = (a_{2 – 1}) + 5
a₂ = a₁ + 5

Substitute 8 for a₁ to find a₂.

a₂ = 8 + 5
a₂ = 13

Repeat this process to find a₃.

a₃ = 13 + 5
a₃ = 18

Repeat this process again until you find a₅.

a₄ = 18 + 5 = 23

a₅ = 23 + 5 = 28

Find the recursive rule for the sequence: 1, 6, 11, …

Locate the first term.

a₁ = 1

Hey, there he is. Now determine how to go from one term to another in the simplest way possible.

1 → 6 by adding 5
6 → 11 by adding 5

So we're adding 5 to each previous term. Rewrite the final rule using braces in terms of a_{n – 1}. Braces still look like this: { }.

{a_n} = (a_{n – 1}) + 5

Find the recursive rule for the sequence: 7, 15, 31, …

Start with the first term.

a₁ = 7

Determine how to go from one term to another in the simplest way possible.

7 → 15 by adding 8
15 → 31 by adding 16

Those are both true, but we need to find a way to go from one term to another that works each and every time. Mistakes will be made. Just keep truckin'.

7 + 8 = 15 but 15 + 8 ≠ 31

Nope, we're not adding 8 to each term. If you get stuck, look at the relationship between the numbers you added or subtracted to get from one term to the next.

8 → 16 by either adding 8 or multiplying by 2

The comparison may help you find a rule that works. Hmm...maybe we're doubling each term and then adding 1?

7 × 2 + 1 = 15

15 × 2 + 1 = 31

Score. Rewrite the final rule using braces in terms of a_{n – 1}.

{a_n} = 2(a_{n – 1}) + 1