EXAMPLE 2 Write rules for sequences

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EXAMPLE 2 Write rules for sequences Describe the pattern, write the next term, and write a rule for the nth term of the sequence (a) –1, –8, –27, –64, . . . and (b) 0, 2, 6, 12, . . . . SOLUTION You can write the terms as (–1)3, (–2)3, (–3)3, (–4)3, . . . . The next term is a5 = (–5)3 = –125. A rule for the nth term is an = 5 (–n)3. a. You can write the terms as 0(1), 1(2), 2(3), 3(4), . . . . The next term is f (5) = 4(5) = 20. A rule for the nth term is f (n) = (n – 1)n. b.

EXAMPLE 3 Solve a multi-step problem Retail Displays You work in a grocery store and are stacking apples in the shape of a square pyramid with 7 layers. Write a rule for the number of apples in each layer. Then graph the sequence. First Layer SOLUTION Make a table showing the number of fruit in the first three layers. Let an represent the number of apples in layer n. STEP 1

EXAMPLE 3 Solve a multi-step problem STEP 2 Write a rule for the number of apples in each layer. From the table, you can see that an = n2. STEP 3 Plot the points (1, 1), (2, 4), (3, 9), . . . , (7, 49). The graph is shown at the right.

GUIDED PRACTICE for Examples 2 and 3 4. For the sequence 3, 8, 15, 24.....describe the pattern,write the next term, graph the first five terms, and write a rule for the nth term. ANSWER You can write the terms as 1 3, 2 4, 3 5, 4 6, . . .; The next term is 35. A rule for the nth term is an = n(n + 2).

GUIDED PRACTICE for Examples 2 and 3 5. WHAT IF? In Example 3, suppose there are 9 layers of apples. How many apples are in the 9th layer? 81 apples ANSWER