Splash Screen

Concept

Find the 20th term of the arithmetic sequence 3, 10, 17, 24, … . Find the nth Term Find the 20th term of the arithmetic sequence 3, 10, 17, 24, … . Step 1 Find the common difference. 24 – 17 = 7 17 – 10 = 7 10 – 3 = 7 So, d = 7. Example 1

an = a1 + (n – 1)d nth term of an arithmetic sequence Find the nth Term Step 2 Find the 20th term. an = a1 + (n – 1)d nth term of an arithmetic sequence a20 = 3 + (20 – 1)7 a1 = 3, d = 7, n = 20 = 3 + 133 or 136 Simplify. Answer: The 20th term of the sequence is 136. Example 1

Find the 17th term of the arithmetic sequence 6, 14, 22, 30, … . B. 140 C. 146 D. 152 Example 1

d = –6 – (–8) or 2; –8 is the first term. Write Equations for the nth Term A. Write an equation for the nth term of the arithmetic sequence below. –8, –6, –4, … d = –6 – (–8) or 2; –8 is the first term. an = a1 + (n – 1)d nth term of an arithmetic sequence an = –8 + (n – 1)2 a1 = –8 and d = 2 an = –8 + (2n – 2) Distributive Property an = 2n – 10 Simplify. Answer: an = 2n – 10 Example 2A

an = a1 + (n – 1)d nth term of an arithmetic sequence Write Equations for the nth Term B. Write an equation for the nth term of the arithmetic sequence below. a6 = 11, d = –11 First, find a1. an = a1 + (n – 1)d nth term of an arithmetic sequence 11 = a1 + (6 – 1)(–11) a6 = 11, n = 6, and d = –11 11 = a1 – 55 Multiply. 66 = a1 Add 55 to each side. Example 2B

Then write the equation. Write Equations for the nth Term Then write the equation. an = a1 + (n – 1)d nth term of an arithmetic sequence an = 66 + (n – 1)(–11) a1 = 66, and d = –11 an = 66 + (–11n + 11) Distributive Property an = –11n + 77 Simplify. Answer: an = –11n + 77 Example 2B

A. Write an equation for the nth term of the arithmetic sequence below A. an = –9n – 21 B. an = 9n – 21 C. an = 9n + 21 D. an = –9n + 21 Example 2A

B. Write an equation for the nth term of the arithmetic sequence below B. Write an equation for the nth term of the arithmetic sequence below. a4 = 45, d = 5 A. an = 5n + 25 B. an = 5n – 20 C. an = 5n + 40 D. an = 5n + 30 Example 2B

Find the arithmetic means in the sequence 21, ___, ___, ___, 45, … . Find Arithmetic Means Find the arithmetic means in the sequence 21, ___, ___, ___, 45, … . Step 1 Since there are three terms between the first and last terms given, there are 3 + 2 or 5 total terms, so n = 5. Step 2 Find d. an = a1 + (n – 1)d Formula for the nth term 45 = 21 + (5 – 1)d n = 5, a1 = 21, a5 = 45 45 = 21 + 4d Distributive Property 24 = 4d Subtract 21 from each side. 6 = d Divide each side by 4. Example 3

Step 3 Use the value of d to find the three arithmetic means. Find Arithmetic Means Step 3 Use the value of d to find the three arithmetic means. 21 27 33 39 45 +6 Answer: The arithmetic means are 27, 33, and 39. Example 3

Find the three arithmetic means between 13 and 25. Example 3

Concept

We need to find n before we can use one of the formulas. Use the Sum Formulas Find the sum 8 + 12 + 16 + … + 80. Step 1 a1 = 8, an = 80, and d = 12 – 8 or 4. We need to find n before we can use one of the formulas. an = a1 + (n – 1)d nth term of an arithmetic sequence 80 = 8 + (n – 1)(4) an = 80, a1 = 8, and d = 4 80 = 4n + 4 Simplify. 19 = n Solve for n. Example 4

Step 2 Use either formula to find Sn. Use the Sum Formulas Step 2 Use either formula to find Sn. Sum formula a1 = 8, n = 19, d = 4 Simplify. Answer: 836 Example 4

Find the sum 5 + 12 + 19 + … + 68. A. 318 B. 327 C. 340 D. 365 Example 4

Step 1 Since you know a1, an, and Sn, use to find n. Find the First Three Terms Find the first three terms of an arithmetic series in which a1 = 14, an = 29, and Sn = 129. Step 1 Since you know a1, an, and Sn, use to find n. Sum formula Sn = 129, a1 = 14, an = 29 Simplify. Divide each side by 43. Example 5

an = a1 + (n – 1)d nth term of an arithmetic sequence Find the First Three Terms Step 2 Find d. an = a1 + (n – 1)d nth term of an arithmetic sequence 29 = 14 + (6 – 1)d an = 29, a1 = 14, n = 6 15 = 5d Subtract 14 from each side. 3 = d Divide each side by 5. Example 5

Step 3 Use d to determine a2 and a3. a2 = 14 + 3 or 17 Find the First Three Terms Step 3 Use d to determine a2 and a3. a2 = 14 + 3 or 17 a3 = 17 + 3 or 20 Answer: The first three terms are 14, 17, and 20. Example 5

Find the first three terms of an arithmetic series in which a1 = 11, an = 31, and Sn = 105. B. 11, 16, 21 C. 11, 17, 23, 30 D. 17, 23, 30, 36 Example 5

Homework P. 670 # 3-12 (x3), 16 – 52 (x4) skip 32

End of the Lesson