1

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Recursive pattern

Vocabulary

Identify and extend patterns using recursion.

Objective

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

In a recursive pattern or recursive sequence, each term is defined using one or more previous terms. For example, the sequence 1, 4, 7, 10, 13, ... can be defined recursively as follows: The first term is 1 and each term after the first is equal to the preceding term plus 3. You can use recursive techniques to identify patterns. The table summarizes the characteristics of four types of patterns.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

You may need to use trial and error when identifying a pattern. If first, second, and third differences are not constant, check for constant ratios.

Helpful Hint!

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion Example 1: Identifying and Extending a Pattern Identify the type of pattern. Then find the next three numbers in the pattern. A. 16, 54, 128, 250, 432

The third differences are constant, so the pattern is cubic.

16 54 128 250 432

+38 +74 +122 +182 +36 +48 +60

+12 +12

Find the first, second, and, if necessary, third differences.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Example 1 Continued Extend the pattern by continuing the sequence of first, second, and third differences.

16 54 128 250 432 686 1,024 1,458

+38 +74 +122 +182 +254 +338 +434

+72 +84 +96

+12 +12

The next three terms in the pattern are 686, 1,024, and 1,458.

2

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Example 1 Continued B. 800, 400, 200, 100, 50, …

Ratios between the terms are constant, so the sequence is exponential. Extend the pattern by continuing the sequence.

Find the ratio between successive terms.

800 400 200 100 50

÷ 2 ÷ 2 ÷ 2 ÷ 2

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Example 1 Continued

800 400 200 100 50 25 12.5 6.25

The next three terms in the pattern are 25, 12.5, and 6.25.

÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 1

Identify the type of pattern. Then find the next three numbers in the pattern. A. 56, 47, 38, 29, 20, … Find the first differences.

56 47 38 29 20 –9 –9 –9 –9

The first differences are constant, so the pattern is linear.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 1 Continued

Extend the pattern by continuing the sequence of first differences.

56 47 38 29 20 11 2 –7

–9 –9 –9 –9 –9 –9 –9

The next three terms in the pattern are 11, 2, and –7.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 1 Continued

B. 1, 8, 27, 64, 125 …

1 8 27 64 125

The third differences are constant, so the pattern is cubic.

Find the first, second, and, if necessary, third differences.

+7 +19 +37 +61 +12 +18 +24

+6 +6

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 1 Continued

Extend the pattern by continuing the sequence of first, second, and third differences.

1 8 27 64 125 216 343 512

The next three terms in the pattern are 216, 343, 512.

+30 +36 +42 +6 +6

+7 +19 +37 +61 +91 +127 +169

3

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Example 2: Identifying a Function A. Determine whether each function is linear, quadratic, or exponential. Check for constant differences in the x-values.

+2 +2 +2 +2

+3 +12 +21 +30

+9 +9 +9

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Example 2 Continued There is a constant change in the x-values. Second differences are constant. The function is a quadratic function.

B. Determine whether each function is linear, quadratic, or exponential.

+4 +4 +4 +4

×2 ×2 ×2 ×2

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

In Example 2, the constant third differences are 24. To extend the pattern, first find each second difference by adding 24 to the previous second difference. Then find each first difference by adding the second difference below to the previous first difference.

Helpful Hint!

Ratios between terms are constant, so the pattern is exponential.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 2 Several ordered pairs that satisfy a function are given. Determine whether the function is linear, quadratic, cubic or exponential. Then find three additional ordered pairs that satisfy the function. A. {0, 1), (1, 3), (2, 9), (3, 19), (4, 33)}

Make a table. Check for a constant change in the x-values. Then find first, second, and third differences of y-values.

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 2 Continued

x 0 1 2 3 4 y 1 3 9 19 33

+1 +1 +1 +1

+2 +6 +10 +14

+4 +4 +4

There is a constant change in the x-values. Second differences are constant, so the function is a quadratic function.

4

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion

Check It Out! Example 2 Continued To find additional ordered pairs, extend the pattern by working backward from the constant second differences.

x 0 1 2 3 4 5 6 7 y 1 3 9 19 33 51 73 99

+1 +1 +1 +1 +1 +1 +1

+2 +6 +10 +14 +18 +22 +26 +4 +4 +4 +4 +4 +4

Three additional ordered pairs that satisfy this function are (5, 51), (6, 73), and (7, 99).

Holt McDougal Algebra 1

9-3-EXT Patterns and Recursion Check It Out! Example 2 Continued

There is a constant change in the x-values. Ratios

between successive y-values are constant at 1 3

Three additional ordered pairs that satisfy this function are

B.

11 , 1 486

13 , 1 1458

13 , 1 1458 , and ,

1, 1 2

3, 1 6

5, 1 18

7, 1 54 9, 1

162 , , , ,