Sequences

1. Sequences Section 14.1

2. Sequences Suppose that a town’s present population of 100,000 is growing by 5% each year. After the first year, the town’s population will be: 100,000 + 0.05(100,000) = 105,000 After the second year, the town’s population will be: 105,000 + 0.05(105,000) = 110,250 After the third year, the town’s population will be: 110,250+ 0.05(110,250) ≈ 115,763

3. Sequences If we continue to calculate, the town’s yearly population can be written as the infinite sequence of numbers 105,000, 110,250, 115,763, …. If we decide to stop calculating after a certain year (say, the fourth year), we obtain the finite sequence 105,000, 110,250, 115,763, 121,551

4. Sequences • An infinite sequence is a function whose domain is the set of natural numbers {1, 2, 3, 4, …} • Ex. 2, 4, 6, 8, … • A finite sequence is a function whose domain is the set of natural numbers {1, 2, 3, 4, …, n}, where n is some natural number. • Ex. 1, -2, 3, -4, 5

5. Writing the terms of a Sequence is known as the general term

6. Writing the terms of a Sequence First term Second term Third term Tenth term

7. Example 1: Write the first three terms of the sequence whose general term is given by . Evaluate , wheren is 1, 2, and 3. Replace n with 1

8. Example 1: Write the first three terms of the sequence whose general term is given by . Evaluate , wheren is 1, 2, and 3. Replace n with 2

9. Example 1: Write the first three terms of the sequence whose general term is given by . Evaluate , wheren is 1, 2, and 3. Replace n with 3

10. OYO: • Write the first four terms of the sequence whose general term is given by .

11. Example 2: If the general term of a sequence is given by , find a. the first term of the sequence

12. Example 2: If the general term of a sequence is given by , find b.

13. Example 2: If the general term of a sequence is given by , find c. the one-hundredth term of the sequence

14. Example 2: If the general term of a sequence is given by , find d.

15. OYO: • If the general term of a sequence is given by , find a. the first term of the sequence b. c. The thirtieth term of the sequence d.

16. Example 3: Finding the general term of a sequence Find a general term of the sequence whose first few terms are given. a. 1, 4, 9, 16, … Terms are all squares.

17. Example 3: Finding the general term of a sequence Find a general term of the sequence whose first few terms are given. b. Terms are all reciprocals.

18. Example 3: Finding the general term of a sequence Find a general term of the sequence whose first few terms are given. c.

19. Example 3: Finding the general term of a sequence Find a general term of the sequence whose first few terms are given. d. Terms double each time.

20. OYO: Find a general term of the sequence whose first few terms are given. a. b. 3, 9, 27, 81, ...

21. Example 4: Application The amount of weight, in pounds, a puppy gains in each month of its first year is modeled by a sequence whose general term is , where nis the number of the month. Write the first five terms of the sequence, and find how much weight the puppy should gain in its fifth month. The puppy should gain 9 pounds in its fifth month.

22. OYO: The value v, in dollars, of an office copier depreciates according to the sequence , where n is the time in years. Find the value of the copier after three years. The copier would only be worth $2022.40 after three years.

23. HOMEWORK • Unit 19 homework page (on the back of your unit plan) # 1 – 5