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**11.2 Arithmetic Sequences & Series**

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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!)

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**Example: Decide whether each sequence is arithmetic.**

5,11,17,23,29,… 11-5=6 17-11=6 23-17=6 29-23=6 Arithmetic (common difference is 6) -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)

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**Rule for an Arithmetic Sequence**

an=a1+(n-1)d

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**Example: Write a rule for the nth term of the sequence 32,47,62,77,…**

Example: 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

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**Example: One term of an arithmetic sequence is a8=50**

Example: One term of an arithmetic sequence is a8=50. The common difference is 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= n-.25 an=48+.25n

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Now graph an=48+.25n. Just like yesterday, remember to graph the ordered pairs of the form (n,an) So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.

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**an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n**

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

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

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**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

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**Example: Consider the arithmetic series 20+18+16+14+… .**

Find the sum of the 1st 25 terms. First find the rule for the nth term. an=22-2n So, a25 = -28 (last term) Find n such that Sn=-760

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**Always choose the positive solution!**

-1520=n( n) -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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Assignment

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