1.3 Arithmetic Sequences

1. 1.3 Arithmetic Sequences Objectives: Identify and graph an arithmetic sequence. Find a common difference. Write an arithmetic sequence recursively and explicitly. Use summation notation. Find the nth term and the nth partial sum of an arithmetic sequence.

2. Example #1 Arithmetic Sequence • Are the following sequences arithmetic? If so, what is the difference between each term and the term preceding it? • {33, 27, 21, 16, 11, 7, 3, …} • {4, 10, 16, 22, 28, 34, …} No, the difference is not constant. Yes, 6 is being added each time. An arithmetic sequence, which is sometimes called an arithmetic progression, is a sequence in which the difference between each term and the preceding term is always constant. In other words, the same number is always added or subtracted.

3. Recursive form of an arithmetic sequence In an arithmetic sequence {un} un = un−1 +d for some constant d and all n ≥ 2. The number d is called the common difference.

4. Example #2Graph of an Arithmetic Sequence • If {un} is an arithmetic sequence with u1 = 2.5 and u2 = 6 as its first two terms, • Find the common difference. • Write the sequence as a recursive function. d = u2− u1 = 6 − 2.5 = 3.5

5. Example #2Graph of an Arithmetic Sequence • If {un} is an arithmetic sequence with u1 = 2.5 and u2 = 6 as its first two terms, • Give the first six terms of the sequence.

6. Example #2Graph of an Arithmetic Sequence • If {un} is an arithmetic sequence with u1 = 2.5 and u2 = 6 as its first two terms, • Graph the sequence. Just as we saw in lesson 1.2, when a sequence is arithmetic, its points form a straight line.

7. Example #3Explicit form of an arithmetic sequence • Confirm that the sequence un = un−1 + 3 with u1 = −5 can also be expressed as un = −5 + (n− 1)∙3 un = un−1 + 3 un = −5 + (n− 1)∙3

8. Explicit form of an arithmetic sequence In an arithmetic sequence {un} with common difference d, un = u1 + (n− 1)d for every n≥ 1. When a sequence is in explicit form, it is not necessary to use the previous term to find the next term.

9. Example #4Explicit form of an arithmetic sequence • Find the nth term of an arithmetic sequence with first term −3 and common difference of 4. • What would be the 150th term?

10. Example #5Finding a term of an arithmetic sequence • What is the 38th term of the arithmetic sequence whose first three terms are 15, 10, and 5? Since the first two terms are 15 and 10, the common difference can be found by subtracting them. d = 15 − 10 = 5

11. Example #6Finding Explicit and recursive formulas • If {un} is an arithmetic sequence with u5 = 22 and u11 = 64, find u1, a recursive formula, and an explicit formula. The explicit formula can be generalized as follows: Where un is the higher term number and up is the lower term number. This general formula can then be used to find the common difference and the first term.

12. Example #6Finding Explicit and recursive formulas • If {un} is an arithmetic sequence with u5 = 22 and u11 = 64, find u1, a recursive formula, and an explicit formula. • Since d = 7 and u1 = −6 the explicit formula can be written. • The recursive formula can also be written. • What would be the 50th term?

13. Summation notation means c1 + c2 + c3 + … + cm The Greek letter sigma is used to denote summations. When sequences are written in summation notation, terms of the sequence are found by plugging numbers into the formula and then summing the results. For the notation above, Ck is the sequence formula, k = 1 tells us to start the sequence by plugging in 1 and the top number m tells us when to stop evaluating and summing the sequence.

14. Example #7Sum of a sequence A.) B.)

15. Example #8calculator computation of a sum • Use a calculator to display the first 7 terms of the sequence and to compute the sum To list the first 7 terms of the sequence: 2nd STAT(LIST)  OPS  5:seq( Enter the information as follows: seq(expression, variable, begin, end, increment (optional)) By scrolling over the remaining digits can be read. They are {1, 7, 13, 19, 25, 31, 37}.

16. Example #8calculator computation of a sum • Use a calculator to display the first 7 terms of the sequence and to compute the sum To compute the sum of a sequence on the graphing calculator: 2nd STAT (LIST)  MATH  5:sum( Enter the information as follows: sum(seq(expression, variable, begin, end)

17. Partial sums of an arithmetic sequence • If {un} is an arithmetic sequence with common difference d, then for each positive integer k, the kth partial sum can be found by using either of the following formulas. Use this formula when the last term is known. Use this formula when the last term is unknown.

18. Example #9Partial sum of a sequence • Find the 14th partial sum of the arithmetic sequence 21, 15, 9, 3, … Since the last term is unknown the second formula would work best.

19. Example #10partial sum of a sequence • Find the sum of all multiples of 4 from 4 to 404. This time the last term is known so the first formula will work best.

20. Example #11Application of partial sums • Larry owns an automobile dealership. He spends $18,000 on advertising during the first year, and he plans to increase his advertising expenditures by $1400 in each subsequent year. How much will Larry spend on advertising during the first 9 years? Once again, since the final term is unknown the second formula will work best.