11.2 Arithmetic Sequences & Series p.659

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!)

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)

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

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

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

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.

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

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

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

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

Always choose the positive solution! -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!

Assignment