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!)

Example: Decide whether each sequence is arithmetic. -10,-6,-2,0,2,6,10,… -10,-6,-2,0,2,6,10,… = = = =4 0--2=2 0--2=2 2-0=2 2-0=2 6-2=4 6-2=4 10-6=4 10-6=4 Not arithmetic (because the differences are not the same) 5,11,17,23,29,… 5,11,17,23,29,… 11-5=6 11-5= = = = = = =6 Arithmetic (common difference is 6) Arithmetic (common difference is 6)

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

Example: Write a rule for the nth term of the sequence 32,47,62,77,…. Then, find a 12. The is a common difference where d=15, therefore the sequence is arithmetic. The is a common difference where d=15, therefore the sequence is arithmetic. Use a n =a 1 +(n-1)d Use a n =a 1 +(n-1)d a n =32+(n-1)(15) a n =32+(n-1)(15) a n =32+15n-15 a n =32+15n-15 a n =15n+17 a n =15n+17 a 12 =15(12) + 17=197

Example: One term of an arithmetic sequence is a 8 =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=a =a1 * Now, use an=a1+(n-1)d to find the rule. an=48.25+(n-1)(.25) an= n-.25 an=48+.25n

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

Example: Two terms of an arithmetic sequence are a 5 =10 and a 30 =110. Write a rule for the nth term. Begin by finding the distance between the terms Begin by finding the distance between the terms 30 – 5 = 25 AND 110 – 10 = /25 = 4 so d = 4 a 5 =a 1 +(5-1)d Now solve for a 1 10=a 1 +4*4 a 1 =-6 Now find the rule a n =a 1 +(n-1)d a n =-6+(n-1)(4) a n =-6 + 4n -4 OR a n =4n - 10

Example (part 2): using the rule a n =-10+4n, write the value of n for which a n =-2. -2=-10+4n8=4n2=n

The 4 th term of an arithmetic sequence is 20 and the 13 th term is 65. Write the first several terms of the sequence = – 4 = 9 45/9 = 5 a n = a 1 + (n-1)d a 13 = a 1 + (13-1)5 65 = a 1 + (12)5 65 = a = a 1

Arithmetic Series The sum of the terms in an arithmetic sequence The sum of the terms in an arithmetic sequence The formula to find the sum of a finite arithmetic series is: The formula to find the sum of a finite arithmetic series is: # of terms 1 st Term Last Term

Example: Consider the arithmetic series …. Find the sum of the 1 st 25 terms. Find the sum of the 1 st 25 terms. First find the 25th term. First find the 25th term. a 25 =20+(25-1)(-2)=-28 a 25 =20+(25-1)(-2)=-28 So, a 25 = -28 So, a 25 = -28

Assignment Worksheet pg 68 and pg 69