Arithmetic Sequences and Series Chapter 8.2

Arithmetic Sequence An Arithmetic Sequence has a constant difference between any two consecutive terms. The Common Difference is denoted by d.

Arithmetic or not… 1.) 7, 10, 13, 16, … 2.) 15, 8, 1, -6, … 3.) 8, 4, 2, 1, … 4.) 15, 9, 3, -3, …

Arithmetic Sequence Rule where an is the nth term, a1 is the first term in the sequence, n is the number of terms, and d is the common difference. Note: You need the beginning number and the common difference to write a rule. 𝑎 𝑛 = 𝑎 1 +(𝑛−1)𝑑

Using the Rule… a1 = 3, d = 5 𝑎 𝑛 = 𝑎 1 +(𝑛−1)𝑑 Write a rule for each sequence. 𝑎 𝑛 = 𝑎 1 +(𝑛−1)𝑑 Ex. 1.) 3, 8, 13, 18, … a1 = 3, d = 5 Find the 20th term. 𝑎 𝑛 =3+(𝑛−1)5 𝑎 𝑛 =3+5𝑛−5 𝑎 𝑛 =5𝑛−2 𝑎 1 =15 𝑑=−7 Ex 2.) 15, 8, 1, -6, …

Write a rule given the common difference and a term in the sequence. 𝑎 19 =−45 𝑑=−3 Find the rule for this sequence. Now use the information to write the rule.

Writing a rule given two terms Now we have too many unknowns. We have the 5th term is -43. And we have the 12th term is -8. Now we have produced 2 equations with 2 unknowns.

Now we can solve for the 2 missing variables using system of equations. d=5, now substitute 5 in for d in one of the original equations to find a1. Now substitute d=5 and a1 = -63 into the Arithmetic Sequence Rule formula.

Sum of an Arithmetic Sequence. ;where n is the number of terms, a1 is the first term of the sequence, an is the last term of the sequence, and S is the sum of the sequence. n = (Top – Bottom) + 1 = (8-1) + 1 = 7 + 1 = 8 terms First term a1 = 2(1) - 3= -1 Last term an = 2(8) - 3= 13

Sum Continued… −1+13 2 S = = = 8(6) = 48

HOMEWORK 5/19/16 PAGE 422 1, 2, 6 – 38 even, 44 – 58 even, 59