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!)

Example: Decide whether each sequence is arithmetic. -10,-6,-2,0,2,6,10,…-10,-6,-2,0,2,6,10,… = = =4-2--6=4 0--2=20--2=2 2-0=22-0=2 6-2=46-2=4 10-6=410-6=4 Not arithmetic (because the differences are not the same) 5,11,17,23,29,…5,11,17,23,29,… 11-5=611-5= =617-11= =623-17= =629-23=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. There is a common difference where d=15, therefore the sequence is arithmetic.There is a common difference where d=15, therefore the sequence is arithmetic. Use a n =a 1 +(n-1)dUse 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 =17+15n a n =17+15n a 12 =17+15(12)=197

Example: One term of an arithmetic sequence is a 8 =50. The common difference is Write a rule for the nth term. Use a n =a 1 +(n-1)d to find the 1 st term!Use a n =a 1 +(n-1)d to find the 1 st term! a 8 =a 1 +(8-1)(.25) 50=a 1 +(7)(.25) 50=a =a 1 * Now, use a n =a 1 +(n-1)d to find the rule. a n =48.25+(n-1)(.25) a n = n-.25 a n =48+.25n

Example: Two terms of an arithmetic sequence are a 5 =10 and a 30 =110. Write a rule for the nth term. Begin by writing 2 equations; one for each term given.Begin by writing 2 equations; one for each term given. a 5 =a 1 +(5-1)d OR 10=a 1 +4d And a 30 =a 1 +(30-1)d OR 110=a 1 +29d Now use the 2 equations to solve for a 1 & d.Now use the 2 equations to solve for a 1 & d. 10=a 1 +4d 10=a 1 +4d 110=a 1 +29d (subtract the equations to cancel a 1 ) -100= -25d So, d=4 and a 1 =-6 (now find the rule) a n =a 1 +(n-1)d a n =-6+(n-1)(4) OR a n =-10+4n

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

Arithmetic Series The sum of the terms in an arithmetic sequenceThe 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 …. A) Find the sum of the 1 st 25 terms.A) Find the sum of the 1 st 25 terms. First find the rule for the nth term.First find the rule for the nth term. a n =22-2na n =22-2n So, a 25 = -28 (last term)So, a 25 = -28 (last term) B)Find n such that S n =-760

-1520=n( n) -1520=-2n 2 +42n 2n 2 -42n-1520=0 n 2 -21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive solution!