Patterns and Recursion 9-3-EXT Lesson Presentation Holt McDougal Algebra 1 Holt Algebra 1

Objective Identify and extend patterns using recursion.

Vocabulary Recursive pattern

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.

You may need to use trial and error when identifying a pattern 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!

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 Find the first, second, and, if necessary, third differences. 16 54 128 250 432 +38 +74 +122 +182 +36 +48 +60 +12 +12 The third differences are constant, so the pattern is cubic.

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.

Example 1 Continued B. 800, 400, 200, 100, 50, … Find the ratio between successive terms. 800 400 200 100 50 ÷ 2 ÷ 2 ÷ 2 ÷ 2 Ratios between the terms are constant, so the sequence is exponential. Extend the pattern by continuing the sequence.

Example 1 Continued 800 400 200 100 50 25 12.5 6.25 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 The next three terms in the pattern are 25, 12.5, and 6.25.

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.

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.

Check It Out! Example 1 Continued B. 1, 8, 27, 64, 125 … Find the first, second, and, if necessary, third differences. 1 8 27 64 125 +7 +19 +37 +61 +12 +18 +24 +6 +6 The third differences are constant, so the pattern is cubic.

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 +7 +19 +37 +61 +91 +127 +169 +30 +36 +42 +6 +6 The next three terms in the pattern are 216, 343, 512.

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

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

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

In Example 2, the constant third differences are 24 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!

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.

Check It Out! Example 2 Continued +1 +1 +1 +1 x 1 2 3 4 y 9 19 33 +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.

Check It Out! Example 2 Continued To find additional ordered pairs, extend the pattern by working backward from the constant second differences. +1 +1 +1 +1 +1 +1 +1 x 1 2 3 4 5 6 7 y 9 19 33 51 73 99 +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).

Check It Out! Example 2 Continued 1, 1 2 3, 6 5, 18 7, 54 9, 162 , B. 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 11 , 1 486 13 , 1 1458 13 , 1 1458 , , and