1. 13.2 – Recursive Definitions Essential Question: How and when is a recursive definition used?

2. Recursive Definition Two parts 1)An initial condition that tells where the sequence starts. ( t 1 = 27 ) 2)A recursive equation (or recursive formula) that tells how any term in the sequence is related to the preceding term.

3. Example t 1 = 5 t n = t n – 1 + 3 Find the first 4 terms.

4. Consider the following sequence: 27, 23, 19, 15, … t 1 = 27 Initial condition Recursion equation t n = -4n + 31 Explicit definition (Arithmetic and Geometric)

5. Example a)Give the first five terms of this sequence. What kind of sequence is it? b)Find a formula for t n in terms of n.

6. Example a. Find first 5 terms. b. Is it arithmetic, geometric, or neither? c. Find t 10. 1. t 1 = 2, t n = 3 t n - 1

7. Ex. Cont. 2. t 1 = 1, t n = t n – 1 + n 3. t 1 = 20, t n = t n – 1 – 3

8. Example Evaluate. Find a recursive definition for this sequence.

9. Example t 1 = 3, t 2 = 5, t n = t n – 1 – t n – 2 a. Find the first 8 terms. b. What will the 1000 th term be?

10. Example The population of a certain county grows as a result of two conditions: 1)The annual population growth is 1% of those already in the country. This growth rate equals the birth rate minus the death rate. 2)20,000 people immigration into the country each year. If the population now is 5,000,000 people, what will the population be in 20 years?