<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

9.9: Lines of Numbers

Difficulty Level: At Grade Created by: CK-12

Lines of Numbers – Identify Patterns Reason Proportionally

Teacher Notes

In Lines of Numbers, students examine repeating patterns, predict the \begin{align*}n^{\text{th}}\end{align*} number in the pattern, and figure out the sum of the first \begin{align*}n\end{align*} numbers in the pattern for various values of \begin{align*}n\end{align*}.

Solutions

\begin{align*}1. \!\\ {\;} \quad \text{a}.\ \text{The}\ 25^{\text{th}}\ \text{number is}\ 1: \!\\ {\;} \qquad \ 25 \div 4\ \text{is}\ 6\ \text{with one left over and}\ 4 \times 6 = 24.\ \text{So the}\ 24^{\text{th}}\ \text{number is}\ 4,\ \text{the last number in} \!\\ {\;} \qquad \ \text{the set. The}\ 25^{\text{th}}\ \text{number is}\ 1,\ \text{the first number in the set}. \!\\ {\;} \quad \text{b}.\ \text{The}\ 75^{\text{th}}\ \text{number is}\ 3: \!\\ {\;} \qquad \ 75 \div 4 = 18\ \text{with} \ 3 \ \text{left over and}\ 4 \times 18 = 72.\ \text{So the}\ 72^{\text{nd}}\ \text{number is}\ 4.\!\\ {\;} \qquad \ \text{The}\ 73^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 74^{\text{th}}\ \text{is}\ 2, \ \text{and the}\ 75^{\text{th}}\ \text{is}\ 3. \!\\ {\;} \quad \text{c}.\ \text{The sum of the first}\ 75\ \text{numbers is}\ 186: \!\\ {\;} \qquad \ \text{The sum of one set of}\ 1,\ 2,\ 3, \ \text{and}\ 4\ \text{is}\ 10.\ \text{From the answer to question b,\ we know that} \!\\ {\;} \qquad \ 18\ \text{sets of four numbers ends with}\ 72.\ \text{The sum of the first 72 numbers is} \!\\ {\;} \qquad \ 10 \times 18,\ \text{or}\ 180.\ \text{The three left over numbers are}\ 1,\ 2,\ \text{and}\ 3,\ \text{and their sum is}\ 6.\ \text{So the} \!\\ {\;} \qquad \ \text{sum of the first}\ 75\ \text{numbers in the pattern is}\ 180 + 6,\ \text{or}\ 186.\!\\ 2. \!\\ {\;} \quad \text{a}.\ \text{The}\ 40^{\text{th}}\ \text{number is}\ 2: \!\\ {\;} \qquad 40 \div 3\ \text{is}\ 13\ \text{with one left over and}\ 3 \times 13 = 39.\ \text{So the}\ 39^{\text{th}}\ \text{number is}\ 4,\ \text{the last number} \!\\ {\;} \qquad \text{on the set. The}\ 40^{\text{th}}\ \text{number is}\ 2,\ \text{the first number in the set}. \!\\ {\;} \quad \text{b}.\ \text{The}\ 110^{\text{th}}\ \text{number is}\ 3: \!\\ {\;} \qquad 110 \div 3 = 36\ \text{with} \ 2 \ \text{left over and}\ 3 \times 36 = 108.\ \text{So the}\ 108^{\text{th}}\ \text{number is}\ 4. \!\\ {\;} \qquad \text{The}\ 109^{\text{th}}\ \text{number will be}\ 2,\ \text{and the}\ 110^{\text{th}}\ \text{is}\ 3. \!\\ {\;} \quad \text{c}.\ \text{The sum of the first}\ 110\ \text{numbers is}\ 329: \!\\ {\;} \qquad \text{The sum of one set of}\ 2,\ 3,\ \text{and}\ 4\ \text{is}\ 9.\ \text{From the answer to question b,\ we know that}\ 36 \!\\ {\;} \qquad \text{sets of three numbers ends with}\ 108.\ \text{The sum of the first}\ 108\ \text{numbers is}\ 9 \!\\ {\;} \qquad \times 36,\ \text{or}\ 324.\ \text{The two left over numbers are}\ 2\ \text{and}\ 3,\ \text{and their sum is}\ 5.\ \text{So the sum of the} \!\\ {\;} \qquad \text{first}\ 110\ \text{numbers in the pattern is}\ 324 + 5,\ \text{or}\ 329.\end{align*}

\begin{align*} 3. \!\\ {\;} \quad \text{a}.\ \text{The}\ 70^{\text{th}}\ \text{number is}\ 3: \!\\ {\;} \qquad \ 70 \div 4\ \text{is}\ 17\ \text{with two left over and}\ 4 \times 17 = 68.\ \text{So the}\ 68^{\text{th}}\ \text{number is}\ 7,\ \text{the last number}\!\\ {\;} \qquad \ \text{in the set. The}\ 69^{\text{th}}\ \text{number is}\ 1,\ \text{and the}\ 70^{\text{th}}\ \text{is}\ 3. \!\\ {\;} \quad \text{b}.\ \text{The}\ 175^{\text{th}}\ \text{number is}\ 5: \!\\ {\;} \qquad \ 175 \div 4 = 43\ \text{with} \ 3 \ \text{left over and}\ 4 \times 43 = 172.\ \text{So the}\ 172^{\text{nd}}\ \text{number is}\ 7. \!\\ {\;} \qquad \ \text{The}\ 173^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 174^{\text{th}}\ \text{is}\ 3,\ \text{and the}\ 175^{\text{th}}\ \text{is}\ 5. \!\\ {\;} \quad \text{c}.\ \text{The sum of the first}\ 175\ \text{numbers is}\ 697: \!\\ {\;} \qquad \text{The sum of one set of}\ 1,\ 3,\ 5,\ \text{and}\ 7\ \text{is}\ 16.\ \text{From the answer to question b, we know that}\!\\ {\;} \qquad 43\ \text{sets of four numbers ends with}\ 172.\ \text{The sum of the first}\ 172\ \text{numbers is}\!\\ {\;} \qquad 16 \times 43,\ \text{or}\ 688.\ \text{The three left over numbers are}\ 1,\ 3,\ \text{and}\ 5,\ \text{and their sum is}\ 9.\ \text{So the} \!\\ {\;} \qquad \text{sum of the first}\ 175\ \text{numbers in the pattern is}\ 688 + 9,\ \text{or}\ 697.\!\\ 4. \!\\ {\;} \quad \text{a}.\ \text{The}\ 47^{\text{th}}\ \text{number is}\ 8: \!\\ {\;} \qquad \ 47 \div 3\ \text{is}\ 15\ \text{with two left over and}\ 3 \times 15 = 45.\ \text{So the}\ 45^{\text{th}}\ \text{number is}\ 9,\ \text{the last number} \!\\ {\;} \qquad \ \text{in the set. The}\ 46^{\text{th}}\ \text{number is}\ 7,\ \text{the first number in the set, and the}\ 47^{\text{th}}\ \text{number is}\ 8. \!\\ {\;} \quad \text{b}.\ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7: \!\\ {\;} \qquad \ 100 \div 3 = 33\ \text{with} \ 1 \ \text{left over and}\ 3 \times 33 = 99.\ \text{So the}\ 99^{\text{th}}\ \text{number is}\ 9. \!\\ {\;} \qquad \ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7. \!\\ {\;} \quad \text{c}.\ \text{The sum of the first}\ 100\ \text{numbers is}\ 809: \!\\ {\;} \qquad \text{The sum of one set of}\ 7,\ 8,\ \text{and}\ 9\ \text{is}\ 24.\ \text{From the answer to question b, we know that}\ 33 \!\\ {\;} \qquad \text{sets of three numbers ends with}\ 99.\ \text{The sum of the first}\ 99\ \text{numbers is}\ 33 \!\\ {\;} \qquad \times 24,\ \text{or}\ 802.\ \text{The}\ 100^{\text{th}}\ \text{number is}\ 7.\ \text{So the sum of the first}\ 100\ \text{numbers in the pattern is}\!\\ {\;} \qquad 802 + 7,\ \text{or}\ 809.\end{align*}

\begin{align*}5. \!\\ {\;} \quad \text{a}.\ \text{The}\ 27^{\text{th}}\ \text{number is}\ 6: \!\\ {\;} \qquad \ 27 \div 4\ \text{is}\ 6\ \text{with three left over and}\ 4 \times 6 = 24.\ \text{So the}\ 24^{\text{th}}\ \text{number is}\ 8,\ \text{the last number on} \!\\ {\;} \qquad \ \text{the set. The}\ 25^{\text{th}}\ \text{number is}\ 4,\ \text{the first number in the set, the}\ 26^{\text{th}}\ \text{number is}\ 4,\ \text{and the}\ 27^{\text{th}} \!\\ {\;} \qquad \ \text{number is}\ 6. \!\\ {\;} \quad \text{b}.\ \text{The}\ 78^{\text{th}}\ \text{number is}\ 4: \!\\ {\;} \qquad \ 78 \div 4 = 19\ \text{with} \ 2 \ \text{left over and}\ 4 \times 19 = 76.\ \text{So the}\ 76^{\text{th}}\ \text{number is}\ 8. \!\\ {\;} \qquad \ \text{The}\ 77^{\text{th}}\ \text{number is}\ 4,\ \text{and the}\ 78^{\text{th}}\ \text{is}\ 4. \!\\ {\;} \quad \text{c}.\ \text{The sum of the first}\ 78\ \text{numbers is}\ 426: \!\\ {\;} \qquad \text{The sum of one set of}\ 4,\ 4,\ 6,\ \text{and}\ 8\ \text{is}\ 22.\ \text{From the answer to question b, we know that} \!\\ {\;} \qquad 19\ \text{sets of four numbers ends with}\ 76.\ \text{The sum of the first} \ 76\ \text{numbers is} \!\\ {\;} \qquad 19 \times 22,\ \text{or}\ 418.\ \text{The two left over numbers are}\ 4\ \text{and}\ 4,\ \text{and their sum is}\ 8.\ \text{So the sum of} \!\\ {\;} \qquad \text{the first}\ 78\ \text{numbers in the pattern is}\ 418 + 8,\ \text{or}\ 426.\end{align*}

\begin{align*}6. \!\\ {\;} \quad 1.\ \text{The}\ 70^{\text{th}}\ \text{number is}\ 3: \!\\ {\;} \qquad \ 70 \div 4\ \text{is}\ 17\ \text{with two left over and}\ 4 \times 17 = 68.\ \text{So the}\ 68^{\text{th}}\ \text{number is}\ 7,\ \text{the last number}\!\\ {\;} \qquad \ \text{in the set. The}\ 69^{\text{th}}\ \text{number is}\ 1,\ \text{and the}\ 70^{\text{th}}\ \text{is}\ 3. \!\\ {\;} \quad 2.\ \text{The}\ 175^{\text{th}}\ \text{number is}\ 5: \!\\ {\;} \qquad \ 175 \div 4 = 43\ \text{with} \ 3 \ \text{left over and}\ 4 \times 43 = 172.\ \text{So the}\ 172^{\text{nd}}\ \text{number is}\ 7. \!\\ {\;} \qquad \text{The}\ 173^{\text{rd}}\ \text{number is}\ 1,\ \text{the}\ 174^{\text{th}}\ \text{is}\ 3,\ \text{and the}\ 175^{\text{th}}\ \text{is}\ 5. \!\\ {\;} \quad 3.\ \text{The sum of the first}\ 175\ \text{numbers is}\ 697: \!\\ {\;} \qquad \text{The sum of one set of}\ 1,\ 3,\ 5,\ \text{and}\ 7\ \text{is}\ 16.\ \text{From the answer to question 2, we know that} \!\\ {\;} \qquad 43\ \text{sets of four numbers ends with}\ 172.\ \text{The sum of the first}\ 172\ \text{numbers} \!\\ {\;} \qquad \text{is}\ 16 \times 43,\ \text{or}\ 688.\ \text{The three left over numbers are}\ 1,\ 3,\ \text{and}\ 5,\ \text{and their sum is}\ 9.\ \text{So the} \!\\ {\;} \qquad \text{sum of the first}\ 175\ \text{numbers in the pattern is}\ 688 + 9,\ \text{or}\ 697.\end{align*}

\begin{align*}& \mathbf{Describe:} && \text{The number pattern keeps repeating the numbers}\ 3,\ 4,\ \text{and}\ 4. \\ \\ & \mathbf{My \ job:} && \text{Figure out the}\ 50^{\text{th}}\ \text{and}\ 200^{\text{th}}\ \text{numbers and the sum of the first}\ 200\ \text{numbers}. \\ \\ & \mathbf{Plan:} && \text{The numbers are in sets of three}:\ 3,\ 4,\ \text{and}\ 4. \ \text{For each question,} \\ & && \bullet \text{Find how many sets of three numbers there are and how many numbers are left over}. \\ & && \bullet \text{Then to find the}\ 50^{\text{th}}\ \text{and the}\ 200^{\text{th}}\ \text{numbers}. \\ & && \bullet \text{To find the sum of the first}\ 200\ \text{numbers,\ multiply the number of sets of three} \\ & && \quad \text{numbers times the sum of the three numbers,\ and add the number left over numbers.} \\ \\ & \mathbf{Solve:} && \text{a}. \quad \text{The}\ 50^{\text{th}}\ \text{number is}\ 4: \\ & && \qquad 50 \div 3\ \text{is}\ 16\ \text{with} \ 2 \ \text{left over and}\ 3 \times 16 = 48. \\ & && \qquad \text{So the}\ 48^{\text{th}}\ \text{number is}\ 4,\ \text{the last number in the set}.\\ & && \qquad \text{The}\ 49^{\text{th}}\ \text{number will be}\ 3,\ \text{and the}\ 50^{\text{th}}\ \text{will be}\ 4.\end{align*}

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

\begin{align*}& \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \text{c}.\quad \text{The sum of the first}\ 200\ \text{numbers is}\ 333: \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad 11\ \text{is the sum of one set of}\ 3,\ 4,\ 4 \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{There are}\ 66\ \text{sets of three numbers ends with}\ 198.\\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad 11 \times 66,\ \text{or}\ 726\ \text{is the sum of the first}\ 198\ \text{numbers}.\ 3 + 4,\ \text{or}\ 7\ \text{is the sum of the} \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{two left over numbers}. \\ & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; && \qquad \text{So},\ 726 + 7,\ \text{or}\ 733\ \text{is the sum of the first}\ 200\ \text{numbers in the pattern}.\end{align*}

\begin{align*}& \mathbf{Check:} \\ & && \text{a}.\quad \text{The}\ 1^{\text{st}},\ 4^{\text{th}},\ 7^{\text{th}},\ \text{and so forth numbers are}\ 3'\text{s}.\ \text{All positions one more than a}\\ & && \qquad \text{multiple of}\ 3\ \text{are}\ 3'\text{s}.\ \text{So the}\ 49^{\text{th}}\ \text{number is}\ 3\ \text{and the}\ 50^{\text{th}}\ \text{is}\ 4. \\ & && \text{b}.\quad 199\ \text{is one more than a multiple of}\ 3,\ \text{so the}\ 199^{\text{th}}\ \text{number is}\ 3\ \text{and the}\ 200^{\text{th}}\ \text{is}\ 4. \\ & && \text{c}.\quad \text{There are}\ 66\ \text{sets of} \ 3 \ \text{in}\ 200,\ \text{with two left over. The sum of the first}\\ & && \qquad 66\ \text{sets of three numbers is}\ 11 \times 66,\ \text{or}\ 726,\ \text{and the sum of the two extra numbers} \\ & && \qquad \text{is}\ 7,\ \text{so the sum is}\ 733.\end{align*}

My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Image Attributions

Show Hide Details
Files can only be attached to the latest version of section
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.9.9
Here