9-2 Arithmetic Sequences & Series

1. 9-2 Arithmetic Sequences & Series

2. Story Time… When another famous mathematician was in first grade, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). Write out the teacher’s request in summation notation, then find the answer (no calculators!) Try to figure out an efficient way!

3. The Story of Little Gauss 1 to 100

4. Find the sum from 3 to 1,000 or

5. Arithmetic Sequence: • The difference between consecutive terms is constant (or the same). • The constant difference is also known as the common difference (d). (It’s also that number that you are adding everytime!)

6. Example 1: Decide whether each sequence is arithmetic. • -10,-6,-2,0,2,6,10,… • -6--10=4 • -2--6=4 • 0--2=2 • 2-0=2 • 6-2=4 • 10-6=4 Not arithmetic (because the differences are not the same) • 5,11,17,23,29,… • 11-5=6 • 17-11=6 • 23-17=6 • 29-23=6 • Arithmetic (commondifference is 6)

7. Ex 2: Find a rule for for the following arithmetic sequence. Will your pattern work for every arithmetic sequence? 5, 8, 11, 14, 17, 20, 23…

8. Rule for an Arithmetic Sequence 2 variables need to be known (or solved for): a1 and d Kind of like in y = mx+b, we need to know m and b an=a1+(n-1)d

9. Example 3: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12. • The is a common difference where d=15, therefore the sequence is arithmetic. • Use an=a1+(n-1)d an=32+(n-1)(15) an=32+15n-15 an=17+15n a12=17+15(12)=197

10. Example 4: One term of an arithmetic sequence is a8=50. The common difference is 0.25. Write a rule for the nth term. • Use an=a1+(n-1)d to find the 1st term! a8=a1+(8-1)(.25) 50=a1+(7)(.25) 50=a1+1.75 48.25=a1 * Now, use an=a1+(n-1)d to find the rule. an=48.25+(n-1)(.25) an=48.25+.25n-.25 an=48+.25n

11. Example 5: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term. • Begin by writing 2 equations; one for each term given. a5=a1+(5-1)d OR 10=a1+4d And a30=a1+(30-1)d OR 110=a1+29d • Now use the 2 equations to solve for a1 & d. 10=a1+4d 110=a1+29d (subtract the equations to cancel a1) -100= -25d So, d=4 and a1=-6 (now find the rule) an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n

12. Example 5(part 2): using the rule an=-10+4n, write the value of n for which an=-2. -2=-10+4n 8=4n 2=n

13. Find a rule for the sum of the following arithmetic series • 2+4+6+8+10+12 • 2+4+6+8+10+12+14+16 • 2+4+6+8+10+12+14+16… • Think of the story of Gauss adding 1 to 100 (12+2)(6/2) = 42 (16+2)(8/2) = 72

14. Arithmetic Series • The sum of the terms in an arithmetic sequence • The formula to find the sum of a finite arithmetic series is: Last Term 1st Term # of terms

15. Example 6: Consider the arithmetic series 20+18+16+14+… . Find n such that Sn=-760 • Find the sum of the 1st 25 terms. • We know the 1st term, we need the 25th term. • an=20+(n-1)(-2) • an=22-2n • So, a25 = -28 (last term)

16. -1520=n(20+22-2n) -1520=-2n2+42n 2n2-42n-1520=0 n2-21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive solution!

17. Partial Sums Your book refers to partial sums of an arithmetic sequence. To find the nth partial sum… simply find the sum of the first n terms. Example: To find the 50th partial sum, find the sum of the first 50 terms.

18. Example 7 Consider a job offer with a starting salary of $32,500 and an annual raise of $2500. Determine the total compensation from the company through the first ten years of employment.

19. Real Life Example

20. H Dub 9-2 Pg. 659 #3-47 odd, 57, 58, 81, 82