A two-step pattern is a sequence where the rule requires two different mathematical operations to get from one term to the next. For example, 'multiply by \(2\) and then add \(1\)'.
How can you try to find the rule for a two-step pattern?
First, check for a simple one-step rule (add, subtract, multiply). If that doesn't work, try combining operations, like multiplication followed by addition or subtraction.
Find the rule for the pattern: \(2, 5, 11, 23, ...\)
The rule is 'multiply by \(2\), then add \(1\)'. \(2 \times 2 + 1 = 5\); \(5 \times 2 + 1 = 11\); \(11 \times 2 + 1 = 23\).
Using the rule 'multiply by \(2\), then add \(1\)', what is the next term after \(23\) in the sequence \(2, 5, 11, 23, ...\)?
The next term is \(47\). Calculation: \(23 \times 2 + 1 = 46 + 1 = 47\).
Find the rule for the pattern: \(3, 7, 15, 31, ...\)
The rule is 'multiply by \(2\), then add \(1\)'. \(3 \times 2 + 1 = 7\); \(7 \times 2 + 1 = 15\); \(15 \times 2 + 1 = 31\).
Find the rule for the pattern: \(4, 7, 13, 25, ...\)
The rule is 'multiply by \(2\), then subtract \(1\)'. \(4 \times 2 - 1 = 7\); \(7 \times 2 - 1 = 13\); \(13 \times 2 - 1 = 25\).
A pattern starts at \(1\) and follows the rule 'multiply by \(3\), then add \(2\)'. What are the first three terms?
The first three terms are \(1\), \(5\), and \(17\). \(1 \times 3 + 2 = 5\); \(5 \times 3 + 2 = 17\).
What is the missing term in the sequence \(2, 8, 26, \_\_, 242\)? The rule has two steps.
The missing term is \(80\). The rule is 'multiply by \(3\), then add \(2\)'. \(26 \times 3 + 2 = 78 + 2 = 80\).
Find the rule for the pattern: \(1, 4, 13, 40, ...\)
The rule is 'multiply by \(3\), then add \(1\)'. \(1 \times 3 + 1 = 4\); \(4 \times 3 + 1 = 13\); \(13 \times 3 + 1 = 40\).
Using the rule 'multiply by \(3\), then subtract \(2\)', find the first three terms if the pattern starts at \(2\).
The first three terms are \(2\), \(4\), and \(10\). \(2 \times 3 - 2 = 4\); \(4 \times 3 - 2 = 10\).
Is it possible for a pattern to alternate between two operations, like 'add \(2\)', then 'multiply by \(3\)'?
Yes, this creates a mixed pattern. For example: \(1, 3, 9, 11, 33, ...\) Here we add \(2\), then multiply by \(3\), then add \(2\), and so on.
Describe the pattern: \(5, 10, 8, 16, 14, ...\)
This is a mixed pattern with an alternating rule: 'multiply by \(2\)', then 'subtract \(2\)'. \(5 \times 2 = 10\); \(10 - 2 = 8\); \(8 \times 2 = 16\); \(16 - 2 = 14\).
