9.5: Lines of Numbers
Look at the pattern below. How would you describe it? Can you answer the questions about the pattern? In this concept, we will practice working with number patterns.
Guidance
In order to examine patterns and answer questions about patterns like the ones above, we can use the problem solving steps to help.
 First, describe what you see in the pattern. What numbers are there? When does it start repeating?
 Second, identify what your job is. In these problems, your job will be to answer the questions.
 Third, make a plan. In these problems, your plan should be to figure out how many times the pattern repeats before reaching the given number.
 Fourth, solve. Answer the questions.
 Fifth, check. Make sure that your answers are correct.
Example A
Look at the pattern below and answer the questions.
Solution:
We can use problem solving steps to help us answer the questions about the pattern.
Describe: The number pattern keeps repeating the numbers 1, 2 3, 4.
My Job: Figure out the 25^{th} and 75^{th} numbers and the sum of the first 75 numbers.
Plan: The numbers are in sets of four: 1, 2, 3, and 4. Find how many sets of four numbers there are and how many numbers are left over to find the 25^{th} and 75^{th} numbers. To find the sum of the first 75 numbers, multiply the number of sets of four numbers by the sum of the four numbers. Then add in the left over numbers.
Solve: a. \begin{align*}25 \div 4\end{align*} is 6 with 1 left over and \begin{align*}4 \times 6 =24\end{align*}. So, the 24^{th} number will be 4 (the last number in the set). Therefore, the 25^{th} number will be 1.

 b. \begin{align*}75 \div 4\end{align*} is 18 with 3 left over and \begin{align*}4 \times 18=72\end{align*}. So, the 72^{nd} number will be 4, the 73^{rd} number will be 1, the 74^{th} number will be 2, and the 75^{th} number will be 3.

 c. 10 is the sum of one set of 1, 2, 3, 4. There are 18 sets of four numbers ending with 72. \begin{align*}10 \times 18=180\end{align*} is the sum of the first 72 numbers. \begin{align*}1+2+3=6\end{align*} is the sum of the three left over numbers. So, \begin{align*}180+6=186\end{align*} is the sum of the first 75 numbers in the pattern.
Check: a. The 1^{st}, 5^{th}, 9^{th}, and so forth numbers are 1s. All positions one more than a multiple of 4 are 1s. So the 25^{th} number is 1.

 b. 73 is one more than a multiple of 4, so the 73^{rd} number is 1, the 74^{th} number is 2, and the 75^{th} number is 3.

 c. There are 18 sets of 4 in 75, with three left over. The sum of the first 18 sets of four numbers is \begin{align*}10\times 18=180\end{align*} and the sum of the three extra numbers is 6, so the total sum is 186.
Example B
Look at the pattern below and answer the questions.
Solution:
We can use problem solving steps to help us answer the questions about the pattern.
Describe: The number pattern keeps repeating the numbers 2, 3, 4.
My Job: Figure out the 40^{th} and 110^{th} numbers and the sum of the first 110 numbers.
Plan: The numbers are in sets of three: 2, 3, and 4. Find how many sets of three numbers there are and how many numbers are left over to find the 40^{th} and 110^{th} numbers. To find the sum of the first 110 numbers, multiply the number of sets of three numbers by the sum of the three numbers. Then add in the left over numbers.
Solve: a. \begin{align*}40 \div 3\end{align*} is 13 with 1 left over and \begin{align*}3 \times 13 =39\end{align*}. So, the 39^{th} number will be 4 (the last number in the set). Therefore, the 40^{th} number will be 2.

 b. \begin{align*}110 \div 3\end{align*} is 36 with 2 left over and \begin{align*}3 \times 36=108\end{align*}. So, the 108^{th} number will be 4. The 109^{th} number will be 2, and the 110^{th} number will be 3.

 c. 9 is the sum of one set of 2, 3, 4. There are 36 sets of three numbers ending with 108. \begin{align*}9 \times 36=324\end{align*} is the sum of the first 108 numbers. \begin{align*}2+3=5\end{align*} is the sum of the two left over numbers. So, \begin{align*}324+5=329\end{align*} is the sum of the first 110 numbers in the pattern.
Check: a. The 1^{st}, 4^{th}, 7^{th}, and so forth numbers are 2s. All positions one more than a multiple of 3 are 2s. So the 40^{th} number is 2.

 b. 109 is one more than a multiple of 3, so the 109^{th} number is 2 and the 110^{th} number is 3.

 c. There are 36 sets of 3 in 110, with two left over. The sum of the first 36 sets of three numbers is \begin{align*}9\times 36=324\end{align*} and the sum of the two extra numbers is 5, so the total sum is 329.
Example C
Look at the pattern below and answer the questions.
Solution:
We can use problem solving steps to help us answer the questions about the pattern.
Describe: The number pattern keeps repeating the numbers 1, 3, 5, 7.
My Job: Figure out the 70^{th} and 175^{th} numbers and the sum of the first 175 numbers.
Plan: The numbers are in sets of four: 1, 3, 5, and 7. Find how many sets of four numbers there are and how many numbers are left over to find the 70^{th} and 175^{th} numbers. To find the sum of the first 175 numbers, multiply the number of sets of four numbers by the sum of the four numbers. Then add in the left over numbers.
Solve: a. \begin{align*}70 \div 4\end{align*} is 17 with 2 left over and \begin{align*}4 \times 17 =68\end{align*}. So, the 68^{th} number will be 7 (the last number in the set). Therefore, the 69^{th} number will be 1, and the 70^{th} number will be 3.

 b. \begin{align*}175 \div 4\end{align*} is 43 with 3 left over and \begin{align*}4 \times 43=172\end{align*}. So, the 172^{nd} number will be 7, the 173^{rd} number will be 1, the 174^{th} number will be 3, and the 175^{th} number will be 5.

 c. 16 is the sum of one set of 1, 3, 5, 7. There are 43 sets of four numbers ending with 172. \begin{align*}16 \times 43=688\end{align*} is the sum of the first 172 numbers. \begin{align*}1+3+5=9\end{align*} is the sum of the three left over numbers. So, \begin{align*}688+9=697\end{align*} is the sum of the first 175 numbers in the pattern.
Check: a. The 1^{st}, 5^{th}, 9^{th}, and so forth numbers are 1s. All positions one more than a multiple of 4 are 1s. So the 69^{th} number is 1 and the 70^{th} number is 3.

 b. 173 is one more than a multiple of 4, so the 173^{rd} number is 1, the 174^{th} number is 3, and the 175^{th} number is 5.

 c. There are 43 sets of 4 in 175, with three left over. The sum of the first 43 sets of four numbers is \begin{align*}16\times 43=688\end{align*} and the sum of the three extra numbers is 9, so the total sum is 697.
Concept Problem Revisited
We can use problem solving steps to help us answer the questions about the pattern.
Describe: The number pattern keeps repeating the numbers 3, 4, and 4.
My Job: Figure out the 50^{th} and 200^{th} numbers and the sum of the first 200 numbers.
Plan: The numbers are in sets of three: 3, 4, and 4. Find how many sets of three numbers there are and how many numbers are left over to find the 50^{th} and 200^{th} numbers. To find the sum of the first 200 numbers, multiply the number of sets of three numbers by the sum of the three numbers. Then add in the left over numbers.
Solve: a. \begin{align*}50 \div 3\end{align*} is 16 with 2 left over and \begin{align*}3 \times 16 =48\end{align*}. So, the 48^{th} number will be 4 (the last number in the set). Therefore, the 49^{th} number will be 3, and the 50^{th} number will be 4.

 b. \begin{align*}200 \div 3\end{align*} is 66 with 2 left over and \begin{align*}3 \times 66=198\end{align*}. So, the 198^{th} number will be 4. The 199^{th} number will be 3, and the 200^{th} number will be 4.

 c. 11 is the sum of one set of 3, 4, 4. There are 66 sets of three numbers ending with 198. \begin{align*}11 \times 66=726\end{align*} is the sum of the first 198 numbers. \begin{align*}3+4=7\end{align*} is the sum of the two left over numbers. So, \begin{align*}726+7=733\end{align*} is the sum of the first 200 numbers in the pattern.
Check: a. The 1^{st}, 4^{th}, 7^{th}, and so forth numbers are 3s. All positions one more than a multiple of 3 are 3s. So the 49^{th} number is 3 and the 50^{th} number is 4.

 b. 199 is one more than a multiple of 3, so the 199^{th} number is 3 and the 200^{th} number is 4.

 c. There are 66 sets of 3 in 200, with two left over. The sum of the first 66 sets of three numbers is \begin{align*}11\times 6=726\end{align*} and the sum of the two extra numbers is 7, so the total sum is 733.
Vocabulary
One type of pattern is when a set of objects repeats over and over. In this concept, we saw patterns of numbers where sets of numbers repeated to form a pattern. With any pattern you should be able to describe the pattern and how to get from one step of the pattern to the next.
Guided Practice
Look at each pattern and answer the questions.
1.
2.
3.
Answers:
1. a. The 47^{th} number is 8. \begin{align*}47\div 3\end{align*} is 15 with two left over. So, the 45^{th} number is 9, the 46^{th} number is 7, and the 47^{th} number is 8.
 b. The 100^{th} number is 7. \begin{align*}100 \div 3 \end{align*} is 33 with 1 left over. So, the 99^{th} number is 9 and the 100^{th} number is 7.
 c. The sum of the first 100 numbers is 809. The sum of one set of 7, 8, 9 is 24. From the answer to question b, we know that 33 sets of three numbers ends with 99. The sum of the first 99 numbers is \begin{align*}33\times 24=802\end{align*}. The 100^{th} number is 7, so the sum of the first 100 numbers in the pattern is \begin{align*}802+7=809\end{align*}.
2. a. The 27^{th} number is 6. \begin{align*}27 \div 4\end{align*} is 6 with three left over. So, the 24^{th} number is 8, the 25^{th} number is 4, the 26^{th} number is 4, and the 27^{th} number is 6.
 b. The 78^{th} number is 4. \begin{align*}78 \div 4\end{align*} is 19 with 2 left over. So, the 76^{th} number is 8, the 77^{th} number is 4, and the 78^{th} number is 4.
 c. The sum of the first 78 numbers is 426. The sum of one set of 4, 4, 6, and 8 is 22. From the answer to question b, we know that 19 sets of four numbers ends with 76. The sum of the first 76 numbers is \begin{align*}19\times 22=418\end{align*}. The two left over numbers are 4 and 4, and their sum is 8. So the sum of the first 78 numbers in the pattern is \begin{align*}418+8=426\end{align*}.
3. a. The 38^{th} number is 3. \begin{align*}38\div 4\end{align*} is 9 with two left over. So, the 36^{th} number is 7, the 37^{th} number is 1, and the 38^{th} number is 3.
 b. The 72^{nd} number is 7. \begin{align*}72 \div 4 \end{align*} is 18 with none left over. So, the 72^{nd} number is 7.
 c. The sum of the first 72 numbers is 288. The sum of one set of 1, 3, 5, 7 is 16. From the answer to question b, we know that 18 sets of four numbers ends with 72. The sum of the first 72 numbers is \begin{align*}16 \times 18=288\end{align*}.
Practice
Look at each pattern and answer the questions.
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Students examine repeating patterns, predict future numbers in the pattern, and figure out the sum of a given number of numbers in the pattern. Students use problem solving steps to help.