2. Understanding 8.1… Use sigma notation to write the sum of

3. Arithmetic Sequences 8.2 JMerrill, 2007 Revised 2008

4. Sequences A Sequence: Usually defined to be a function Domain is the set of positive integers Arithmetic sequence graphs are linear (usually)

5. Sequences SEQUENCE - a set of numbers, called terms, arranged in a particular order. There are two basic types: Arithmetic Geometric This unit deals with arithmetic sequences

6. Arithmetic Sequences ARITHMETIC - the difference of any two consecutive terms is constant. In order to find the difference, you MUST pick one term and subtract the preceding term You MUST check more than 1 pair of terms! 2,6,10,14,18……… difference = 4 17,10,3,-4,-11,-18……. difference = -7 a, a+d, a+2d, a+3d…………. difference = d Are you Ready???

7. The difference of 8, 3, -2, -7… is 5 True or False?

8. The difference of 23, 17, 11, 5, is -6 True or False?

9. Formulas for the n th term of a Sequence Arithmetic:an = = = = a1 + (n-1)d To get the nth term, start with the 1st term and add the difference (n-1) times n = THE TERM NUMBER

10. Example Find a formula for an and sketch the graph for the sequence 1, 4, 7, 10... Arithmetic or Geometric? d = ? an = = = = a1 + (n-1)d an = = = = 1 + (n-1)3 an = = = = 1 + 3n-3 an = = = = -2 + 3n 3 n = THE TERM NUMBER

11. Example Find the given term of the arithmetic sequence if a1 = 15, a2 =21, find a20 d = an = = = = a1 + (n-1)d a20 = = = = 15 + (19)6 a20 = = = = 15 + 114 a20 = = = = 129 6

12. You Do Find a formula for the nth term of an arithmetic sequence whose common difference is 3 and whose first term is 2 a n = 3n - 1

13. Last Example The 4th term of an arithmetic sequence is 20, and the 13th term is 65. Write the first 3 terms of the sequence? 1st use the equation: an = = = = a1 + (n-1)d a4 = = = = a1 + (4-1)da13 = = = = a1 + (13-1)d 20 = = = = a1 + 3d65 = = = = a1 + 12d 20 – 3d = a165 – 12d = a1 20 – 3d = 65 – 12d; d = 5

14. Knowing that d = 5 and the 4th term is 20, we can subtract 5 each time and know that the sequence is 5, 10, 15, 20… If we had been asked to find the equation (and we couldn’t figure out that the 1st term was 5)… 20 = = = = a1 + 3d 20 = = = = a1 + 3(5) 5 = a1 So, an = = = = 5 + (n-1)5 and an = = = = 5n

15. Understanding Problem Write the 1 st 5 terms of the sequence. If the sequence is arithmetic, find the common difference.

16. Sum of a Finite Arithmetic Series The sum of the 1 st n terms of an arithmetic series is You can see that you need the first term and the nth term.

17. Example Find the sum of the 1 st 25 terms of the arithmetic series 11 + 14 + 17 + 20 + … Step 1: Find the 25 th term:

18. Example Find the sum of the cubes of the first twenty positive integers. So, we want S 20 = 1 3 + 2 3 + 3 3 + …+ 20 3 a 1 = 1 a 20 = 20 3 = 8000

19. You Do Find the 150 th partial sum of the arithmetic sequence 5, 16, 27, 38, 49 Can you do it? 123,675

20. Last Example An auditorium has 20 rows of seats. There are 20 seats in the 1st row, 21 seats in the 2nd row, 22 seats in the 3rd row, and so on. How many seats are there in all 20 rows? a1 = 20 a2 = 21 a3 = 22 d = 1 a n = a 1 + (n-1)d a 20 = 20 + 19 a 20 = 39

21. Last Problem Find the partial sum of the following problem WITHOUT a calculator (use formula) 218,625