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9.2 Arithmetic Sequences & Series

9.2 Arithmetic Sequences & Series. 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!). -10,-6,-2,0,2,6,10,… -6--10=4 -2--6=4

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9.2 Arithmetic Sequences & Series

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  1. 9.2 Arithmetic Sequences & Series

  2. 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!)

  3. -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 (common difference is 6) Example: Decide whether each sequence is arithmetic.

  4. Rule for an Arithmetic Sequence an=a1+(n-1)d

  5. Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12. • There 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

  6. Example: 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

  7. Example: 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

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

  9. 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

  10. A) Find the sum of the 1st 25 terms. First find the rule for the nth term. an=22-2n So, a25 = -28 (last term) B)Find n such that Sn=-760 Example: Consider the arithmetic series 20+18+16+14+… .

  11. -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!

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