2Arithmetic 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!)
3Example: Decide whether each sequence is arithmetic. 5,11,17,23,29,…11-5=617-11=623-17=629-23=6Arithmetic (common difference is 6)-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)
5Example: 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)dan=32+(n-1)(15)an=32+15n-15an=17+15na12=17+15(12)=197
6Example: 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+1.7548.25=a1* Now, use an=a1+(n-1)d to find the rule.an=48.25+(n-1)(.25)an= n-.25an=48+.25n
7Now 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.
8an=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+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= -25dSo, d=4 and a1=-6 (now find the rule)an=a1+(n-1)dan=-6+(n-1)(4) OR an=-10+4n
9Example (part 2): using the rule an=-10+4n, write the value of n for which an=-2.
10Arithmetic Series The sum of the terms in an arithmetic sequence The formula to find the sum of a finite arithmetic series is:Last Term1st Term# of terms
11Example: 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=22-2nSo, a25 = -28 (last term)Find n such that Sn=-760
12Always choose the positive solution! -1520=n( n)-1520=-2n2+42n2n2-42n-1520=0n2-21n-760=0(n-40)(n+19)=0n=40 or n=-19Always choose the positive solution!