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8.9: Lines of Numbers

Difficulty Level: At Grade Created by: CK-12
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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*}nth number in the pattern, and figure out the sum of the first \begin{align*}n\end{align*}n numbers in the pattern for various values of \begin{align*}n\end{align*}n.

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*}1.a. The 25th number is 1: 25÷4 is 6 with one left over and 4×6=24. So the 24th number is 4, the last number in the set. The 25th number is 1, the first number in the set.b. The 75th number is 3: 75÷4=18 with 3 left over and 4×18=72. So the 72nd number is 4. The 73rd number is 1, the 74th is 2, and the 75th is 3.c. The sum of the first 75 numbers is 186: The sum of one set of 1, 2, 3, and 4 is 10. 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 10×18, or 180. The three left over numbers are 1, 2, and 3, and their sum is 6. So the sum of the first 75 numbers in the pattern is 180+6, or 186.2.a. The 40th number is 2:40÷3 is 13 with one left over and 3×13=39. So the 39th number is 4, the last numberon the set. The 40th number is 2, the first number in the set.b. The 110th number is 3:110÷3=36 with 2 left over and 3×36=108. So the 108th number is 4.The 109th number will be 2, and the 110th is 3.c. The sum of the first 110 numbers is 329:The sum of one set of 2, 3, and 4 is 9. From the answer to question b,\ we know that 36sets of three numbers ends with 108. The sum of the first 108 numbers is 9×36, or 324. The two left over numbers are 2 and 3, and their sum is 5. So the sum of thefirst 110 numbers in the pattern is 324+5, or 329.

\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*}3.a. The 70th number is 3: 70÷4 is 17 with two left over and 4×17=68. So the 68th number is 7, the last number in the set. The 69th number is 1, and the 70th is 3.b. The 175th number is 5: 175÷4=43 with 3 left over and 4×43=172. So the 172nd number is 7. The 173rd number is 1, the 174th is 3, and the 175th is 5.c. The sum of the first 175 numbers is 697:The sum of one set of 1, 3, 5, and 7 is 16. From the answer to question b, we know that43 sets of four numbers ends with 172. The sum of the first 172 numbers is16×43, or 688. The three left over numbers are 1, 3, and 5, and their sum is 9. So thesum of the first 175 numbers in the pattern is 688+9, or 697.4.a. The 47th number is 8: 47÷3 is 15 with two left over and 3×15=45. So the 45th number is 9, the last number in the set. The 46th number is 7, the first number in the set, and the 47th number is 8.b. The 100th number is 7: 100÷3=33 with 1 left over and 3×33=99. So the 99th number is 9. The 100th number is 7.c. The sum of the first 100 numbers is 809:The sum of one set of 7, 8, and 9 is 24. From the answer to question b, we know that 33sets of three numbers ends with 99. The sum of the first 99 numbers is 33×24, or 802. The 100th number is 7. So the sum of the first 100 numbers in the pattern is802+7, or 809.

\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*}

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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.9.9
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