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Given the explicit rule: {a_{n}} = 2n – 5, find a_{10}.

a_{10} = 2(10) – 5

Substitute 10 for n in your rule.

a_{10} = 20 – 5a_{10} = 15 Simplify and solve for a_{10}.

Example 2

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

a_{1} = 2 a_{2} = 4 a_{3} = 6

Match each term with their term number, n.

1 to 2, 2 to 4, 3 to 6

Look at each term number next to its corresponding term.

1 × 2 = 2, 2 × 2 = 4, 3 × 2 = 6

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

{a_{n}} = 2n

Rewrite the final rule using braces in terms of n. Braces look like this: { }.

Example 3

Find a_{5} for the sequence: 1, 5, 9, …

1 → 5 by adding 4 5 → 9 by adding 4

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

9 + 4 = 13 13 + 4 = 17

If the term number you need is small enough, just apply the rule and go for it.

a_{5} = 17

Example 4

Find a_{20} for the sequence 2, 4, 8, …

a_{1} = 2 a_{2} = 4 a_{3} = 8 so a_{n} = 2^{n} The term number here is big. So find a rule that allows you to just plug in the term number you need for n.