Given the explicit rule {a_n} = 2n – 5, find a₁₀.

We want the tenth term in the sequence, so all we've gotta do is substitute 10 for n in our rule.

a₁₀ = 2(10) – 5

Simplify and solve for a₁₀.

a₁₀ = 20 – 5

a₁₀ = 15

Find the explicit rule for the sequence: 2, 4, 6, 8, …

First we'll match each term with its term number, n.

a₁ = 2
a₂ = 4
a₃ = 6
a₄ = 8

Hmm...we're definitely adding 2 to each term that came before, but we want an explicit rule, not a recursive one. Let's think multiplication. Take a look at each term number next to its corresponding term.

1 to 2

2 to 4

3 to 6

4 to 8

Find a way to relate the term number to each term along the way.

1 × 2 = 2
2 × 2 = 4
3 × 2 = 6
4 × 2 = 8

There we go. It looks like we're multiplying the term number by 2 every time. Now we can rewrite the final rule using braces in terms of n. Braces look like this: { }.

{a_n} = 2n

Test 'er out real quick on the fourth term (or whichever one you want):

a₄ = 2(4) = 8

Yep, it works.

Find a₅ for the sequence: 1, 5, 9, …

The term number we want is pretty close to what we were given. So figure out a consistent way to get from one term to another.

1 → 5 by adding 4
5 → 9 by adding 4

We're adding 4 every time. If the term number you need is small enough, just apply the rule and go for it. In this case, we can just keep adding 4 till we get to the fifth term.

a₄ = 9 + 4 = 13
a₅ = 13 + 4 = 17

Find a₂₀ for the sequence 2, 4, 8, …

The term number here is pretty big. So find a rule that allows you to just plug in the term number you need for n.

a₁ = 2
a₂ = 4
a₃ = 8

We need an explicit rule. Since we're multiplying by 2 every time, we can use an exponent.

{a_n} = 2ⁿ

Plug in the term number you need for n. Better grab a calculator for this one.

a₂₀ = 2²⁰ = 1,048,576