Two number patterns both start at \(2\). Pattern A rule: add \(3\). Pattern B rule: add \(5\). Write the first five terms of each pattern, including the starting term.

Two number patterns start with the values shown. The first pattern follows the rule “add \(2\).” The second follows the rule “add \(4\).” Complete position \(2\). <table><tr><th>Position</th><th>First pattern</th><th>Second pattern</th></tr><tr><td>1</td><td>\(3\)</td><td>\(8\)</td></tr><tr><td>2</td><td>\(\square\)</td><td>\(\square\)</td></tr></table>

Here are two completed patterns. First pattern: \(3, 6, 9, 12\) Second pattern: \(5, 9, 13, 17\) a) Which two terms are in position \(3\)? b) At which position do \(12\) and \(17\) correspond?

Pattern A is made by adding one more \(6\) on the right: \(6,66,666,6666\). Pattern B follows this rule: start with \(72\), then insert one more \(9\) just before the final \(2\). a) Write the first four terms of Pattern B. b) Check each pair by multiplying the Pattern A term by \(12\). What relationship do you see?

Pattern A starts at \(13\) and adds \(13\): \(13,26,39,52,65\). Pattern B starts at \(1001\) and adds \(1001\). a) Write the first five terms of Pattern B. b) Compare corresponding terms. What multiplication rule connects Pattern A to Pattern B?

Pattern A is made by adding one more \(9\) on the right: \(9,99,999,9999,99{,}999\). Pattern B follows this rule: start with \(81\), then insert one more \(9\) just before the final \(1\). a) Write the first five terms of Pattern B. b) Check each pair by multiplying the Pattern A term by \(9\). What relationship do you see?

Pattern A starts at \(4\) and follows the rule “add \(6\).” Pattern B starts at \(9\) and also follows the rule “add \(6\).” Complete the table and describe the relationship between corresponding terms. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(4\)</td><td>\(9\)</td></tr><tr><td>2</td><td>\(10\)</td><td></td></tr><tr><td>3</td><td>\(16\)</td><td>\(21\)</td></tr><tr><td>4</td><td>\(22\)</td><td></td></tr><tr><td>5</td><td>\(28\)</td><td>\(33\)</td></tr></table>

Pattern A starts at \(3\) and follows the rule “add \(3\).” Pattern B starts at \(6\) and follows the rule “add \(6\).” Complete the table and state how each Pattern B term is related to the corresponding Pattern A term. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(3\)</td><td>\(6\)</td></tr><tr><td>2</td><td></td><td>\(12\)</td></tr><tr><td>3</td><td>\(9\)</td><td></td></tr><tr><td>4</td><td></td><td>\(24\)</td></tr><tr><td>5</td><td>\(15\)</td><td>\(30\)</td></tr></table>

Complete both patterns for four positions. First pattern: start at \(4\), then add \(2\) each time. Second pattern: start at \(7\), then add \(4\) each time. After you complete them, subtract the first-pattern term from the second-pattern term at each position. How does this difference change from one position to the next?

Two number patterns begin at \(40\). Pattern A: add \(1\) each time: \(40,41,42,43,44\). Pattern B: subtract \(1\) each time: \(40,39,38,37,36\). Multiply corresponding terms. How do the products change as the two patterns move away from \(40\)?

Pattern A starts at \(7\) and adds \(7\). Pattern B starts at \(111{,}111\) and adds \(111{,}111\). a) Write the first five terms of both patterns. b) Explain how multiplying each Pattern A term by \(15{,}873\) produces the corresponding Pattern B term.

Pattern A starts at \(2\) and adds \(2\): \(2,4,6,8,10,12\). Pattern B starts at \(1616\) and adds \(1616\). a) Write the first six terms of Pattern B. b) Explain why each Pattern B term equals \(808\) times the corresponding Pattern A term.

Pattern A starts at \(10\) and follows the rule “add \(5\).” Pattern B starts at \(4\) and follows the rule “add \(8\).” The last row of the table is missing its position and Pattern A term. Complete that row. <table><tr><th>Position</th><th>Pattern A</th><th>Pattern B</th></tr><tr><td>1</td><td>\(10\)</td><td>\(4\)</td></tr><tr><td>2</td><td>\(15\)</td><td>\(12\)</td></tr><tr><td>3</td><td>\(20\)</td><td>\(20\)</td></tr><tr><td></td><td></td><td>\(28\)</td></tr></table>

Pattern A starts at \(5\) and follows the rule “add \(4\).” Pattern B starts at \(8\) and follows the rule “add \(7\).” Paolo says, “The first terms differ by \(3\), so every pair of corresponding terms will differ by \(3\).” Is Paolo correct? Use the first four pairs of terms to explain.

Pattern A starts at \(6\) and follows the rule “add \(4\).” Pattern B starts at \(15\) and follows the rule “add \(7\).” At the first position, Pattern B is \(9\) greater than Pattern A. Without writing a variable formula, determine whether the difference between corresponding terms will ever be \(30\). If it will, state the position and explain why.

Pattern A starts at \(4\) and follows the rule “add \(3\).” Pattern B starts at \(10\) and follows the rule “add \(6\).” Generate the first four terms of each pattern. Describe a relationship between corresponding terms, and explain why that relationship continues as both patterns grow.

Pattern A starts at \(3\) and follows the rule “add \(2\).” Pattern B starts at \(7\). The relationship should be: each Pattern B term is \(2\) less than \(3\) times the corresponding Pattern A term. What add rule must Pattern B follow so that this relationship continues? Generate the first four terms of both patterns to verify your rule, and explain why it works.