Warm-Up Write the first five terms of an = 4n + 2 a1 = 4(1) + 2 = 6 1st Term a2 = 4(2) + 2 = 10 2nd Term a3 = 4(3) + 2 = 14 3rd Term a4 = 4(4) + 2 = 18 4th Term a5 = 4(5) + 2 = 22 5th Term

Analyze Arithmetic Sequences and Series Section 12-2 Analyze Arithmetic Sequences and Series

Vocabulary Arithmetic Sequence – The difference of consecutive terms is constant. Common Difference – The constant difference, denoted by d. Arithmetic Series – The expression formed by adding the terms of an arithmetic sequence.

Rule for an Arithmetic Sequence The nth term of an arithmetic sequence with first term a1 and the common difference d is given by: an = a1 + (n – 1)d The Sum of a Finite Arithmetic Series The sum of the first n terms of an arithmetic series is: a1 + an 2 Sn is the mean of the 1st and nth terms, multiplied by the number of terms. Sn = n

Example 1 Tell whether the sequence is arithmetic. b.) -8, -4, 0, 6, 12, … a2 – a1 = 13 – 7 = 6 YES a3 – a2 = 19 – 13 = 6 a4 – a3 = 25 – 19 = 6 a2 – a1 = -4 – (-8) = 4 NO a3 – a2 = 0 – (-4) = 4 a4 – a3 = 6 – 0 = 6

Example 2 Write a rule for the nth term of the sequence. Then find a20. a.) -7, -10, -13, -16, … an = a1 + (n – 1)d a1 = -7 d = -3 an = -7 + (n – 1)(-3) an = -7 - 3n + 3 Rule: an = -4 - 3n a20 = -4 - 3(20) a20 = -64

Example 2 - continued Write a rule for the nth term of the sequence. Then find a20. b.) 59, 68, 77, 86, … an = a1 + (n – 1)d a1 = 59 d = 9 an = 59 + (n – 1)(9) an = 59 + 9n – 9 Rule: an = 50 + 9n a20 = 50 + 9(20) a20 = 230

Example 3 One term of an arithmetic sequence is a27 = 263. The common difference is d = 11. a.) Write the rule for the nth term. an = a1 + (n – 1)d a27 = a1 + (27 – 1)(11) 263 = a1 + 286 a1 = -23 an = -23 + (n – 1)11 an = -23 + 11n – 11 Rule: an = -34 + 11n

Example 3 - continued b.) Graph the first 6 terms of the sequence. Create a table of values for the sequence. Rule: an = -34 + 11n an n 1 2 3 4 5 6 -23 a1 = -23 -12 a2 = -34 + 11(2) -1 a3 = -34 + 11(3) 10 a4 = -34 + 11(4) 21 a5 = -34 + 11(5) 32 a6 = -34 + 11(6)

Example 3 - continued b.) Graph the first 6 terms of the sequence. an 30 25 an n 1 2 3 4 5 6 20 -23 15 -12 10 5 -1 -1 1 2 3 4 5 6 10 -5 21 -10 32 -15 -20 -25 -30

Example 4 Two terms of an arithmetic sequence are a10 = 148 and a44 = 556. Find a rule for the nth term. Step 1: Write a system of equations using an = a1 + (n – 1)d and substitute 44 for n in equation 1 and then 10 for n in equation 2. a44 = a1 + (44 – 1)d 556 = a1 + 43d a10 = a1 + (10 – 1)d 148 = a1 + 9d 408 = 34d Step 2: Solve the system. d = 12 Step 3: Substitute d into equation 1. 556 = a1 + 43(12) a1 = 40

Example 4 - continued Two terms of an arithmetic sequence are a10 = 148 and a44 = 556. Find a rule for the nth term. Step 4: Find a rule for an. an = a1 + (n – 1)d a1 = 40 d = 12 an = 40 + (n – 1)(12) an = 40 + 12n – 12 an = 28 + 12n

Example 5 ∑ Find the sum of the arithmetic series a1 = -2 + 4i 28 ∑ (-2 + 4i) i = 1 a1 = -2 + 4i Step 1: Identify the first term. a1 = -2 + 4(1) = 2 Step 2: Identify the last term. a28 = -2 + 4i a28 = -2 + 4(28) = 110 Step 3: Write the rule for S28 , substitute 2 for a1 and 110 for a28. S28 = 2 + 110 2 28 Sn = a1 + an 2 n S28 = 1568

Homework Section 12-2 Page 806 - 807 5 – 10, 15 – 18, 23 – 25, 29, 31 – 33, 39, 40 – 43, 46, 55 – 57