amp Series Story Time When another famous mathematician was in first grade his teacher asked the class to add up the numbers one through a hundred 123 etc all the way up to 100 Write out the teachers request in summation notation then find the answer no calculators Try to fig ID: 1002476
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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 Gauss1 to 100
4. Find the sum from 3 to 1,000or
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=40--2=22-0=26-2=410-6=4Not arithmetic (because the differences are not the same)5,11,17,23,29,…11-5=617-11=623-17=629-23=6Arithmetic (common difference is 6)
7. Ex 2: Find a rule for for the following arithmetic sequence.5, 8, 11, 14, 17, 20, 23…Will your pattern work for every arithmetic sequence?
8. Rule for an Arithmetic Sequencean=a1+(n-1)d2 variables need to be known (or solved for): a1 and dKind of like in y = mx+b, we need to know m and b an= d(n-1)+a1
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+15na12=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-.25an=48+.25nThis is like being given a slope and a (x,y) coordinate. We need to find the “b”!
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+4dAnda30=a1+(30-1)d OR 110=a1+29dNow use the 2 equations to solve for a1 & d. 10=a1+4d110=a1+29d (subtract the equations to cancel a1)-100= -25d So, d=4 and a1=-6 (now find the rule)an=a1+(n-1)dan=-6+(n-1)(4) OR an=-10+4nThis is like being given 2 coordinates. We have to find the “slope” and the “b”
12. Example 5(part 2): using the rule an=-10+4n, write the value of n for which an=-2.-2=-10+4n8=4n2=n
13. Find a rule for the sum of the following arithmetic series2+4+6+8+10+122+4+6+8+10+12+14+162+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 SeriesThe sum of the terms in an arithmetic sequenceThe formula to find the sum of a finite arithmetic series is:# of terms1st TermLast Term
15. Example 6: Consider the arithmetic series 20+18+16+14+… .Find the sum of the 1st 25 terms.We know the 1st term, we need the 25th term.an=20+(n-1)(-2)an=22-2nSo, a25 = -28 (last term)Find n such that Sn=-760
16. -1520=n(20+22-2n)-1520=-2n2+42n2n2-42n-1520=0n2-21n-760=0(n-40)(n+19)=0n=40 or n=-19Always choose the positive solution!
17. Partial SumsYour 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 7Consider 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 Dub9-2 Pg. 659 #3-47 odd, 57, 58, 81, 82