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9.10: Lines of Numbers Extra for Experts

Difficulty Level: At Grade Created by: CK-12

Extras for Experts: Lines of Numbers - Identify Patterns and Reason Proportionally

Solutions

1.a. The 50th number is 1: 50÷6 is 8 with 2 left over and 8×6=48. So the 48th number is 3, the last number in the set. The 50th number is 2, the second number in the set.b. The 100th number is 3: 100÷6=16 with 4 left over and 6×16=96. So the 96th number is 3. Then the 100th number is 3.c. The sum of the first 100 numbers is 232: The sum of one set of 1, 2, 2, 3, 3, and 3 is 14. From the answer to question b,\ we know that 16 sets of six numbers ends with 96. The sum of the first 96 numbers is 16×14, or 224. The four left over numbers are 1, 2, 2, and 3, and their sum is 8. So the sum of the first 100 numbers in the pattern is 224+8, or 232.2.a. The 70th number is 4:70÷4 is 17 with two left over and 4×17=68. So the 68th number is 6, the last numberon the set. The 70th number is 4, the second number in the set.b. The 125th number is 3:125÷4=31 with one left over and 4×31=124. So the 124th number is 6.The 125th number will be 3.c. The sum of the first 125 numbers is 561:The sum of one set of 3, 4, 5, and 6 is 18. From the answer to question b,\ we know that 31sets of three numbers ends with 124. The sum of the first 124 numbers is 18×31,or 558. The left over numbers is 3. So the sum of the first 125 numbers in the pattern is558+3, or 561.

3.a. The 67th number is 3: 67÷5 is 13 with two left over and 5×13=65. So the 65th number is 9, the last number in the set. The 66th number is 1, and the 67th is 3.b. The 129th number is 7: 129÷5=25 with 4 left over and 5×25=125. So the 125th number is 7. The 129th is 7, the fourth number in the set.c. The sum of the first 129 numbers is 641:The sum of one set of 1, 3, 5, 7, and 9 is 25. From the answer to question b, we know that25 sets of four numbers ends with 125. The sum of the first 125 numbers is25×25, or 625. The four left over numbers are 1, 3, 5, and 7, and their sum is 16. So thesum of the first 129 numbers in the pattern is 625+16, or 641.

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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.9.10

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