1054 CHAPTER 11 Further Topics in Algebra To find a1, use the formula for an. an = a1 + 1n - 12d Formula for an -16 = a1 + 18 - 121-32 Let an = -16, n = 8, and d = -3. -16 = a1 - 21 Simplify. a1 = 5 Add 21 and interchange sides. S Now Try Exercise 33. To determine the characteristics of the graph of an arithmetic sequence, start by rewriting the formula for the nth term. an = a1 + 1n - 12d Formula for the nth term an = a1 + nd - d Distributive property an = dn + 1a1 - d2 Commutative and associative properties an = dn + c Let c = a1 - d. The points in the graph of an arithmetic sequence are determined by an = dn + c, where n is a natural number. Thus, the discrete points on the graph of the sequence must lie on the continuous linear graph y = dx + c. Slope y-value of the y-intercept For example, the sequence an shown in Figure 7(a) is an arithmetic sequence because the points that make up its graph are collinear—that is, lie on a line. The slope determined by these points is 2, so the common difference d equals 2. On the other hand, the sequence bn shown in Figure 7(b) is not an arithmetic sequence because the points are not collinear. EXAMPLE 6 Finding the n thTerm from a Graph Write a formula for the nth term of the finite arithmetic sequence an shown in Figure 8. Then state the domain and range of the sequence. SOLUTION The points in Figure 8 lie on a line, so the sequence is arithmetic. The dashed line in Figure 9 has slope -0.5 and y-intercept 10, 42, so its equation is y = -0.5x + 4. The nth term of this sequence is defined by an = -0.5n + 4. 0 1 2 3 4 5 6 7 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n an Figure 8 0 1 2 3 4 5 6 7 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n an Figure 9 0 1 2 3 4 5 6 7 2 4 6 8 10 12 14 n an (a) 0 1 2 3 4 5 6 7 2 4 6 8 10 12 14 n bn (b) Figure 7
RkJQdWJsaXNoZXIy NjM5ODQ=