Difference between revisions of "2002 AMC 12B Problems/Problem 7"
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+ | {{duplicate|[[2002 AMC 12B Problems|2002 AMC 12B #7]] and [[2002 AMC 10B Problems|2002 AMC 10B #11]]}} | ||
== Problem == | == Problem == | ||
The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of their [[perfect square|squares]]? | The product of three consecutive positive integers is <math>8</math> times their sum. What is the sum of their [[perfect square|squares]]? | ||
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\qquad\mathrm{(E)}\ 194</math> | \qquad\mathrm{(E)}\ 194</math> | ||
== Solution == | == Solution == | ||
− | Let the three consecutive integers be <math> | + | Let the three consecutive positive integers be <math>a-1</math>, <math>a</math>, and <math>a+1</math>. Since the mean is <math>a</math>, the sum of the integers is <math>3a</math>. So <math>8</math> times the sum is just <math>24a</math>. With this, we now know that <math>a(a-1)(a+1)=24a\Rightarrow(a-1)(a+1)=24</math>. <math>24=4\times6</math>, so <math>a=5</math>. Hence, the sum of the squares is <math>4^2+5^2+6^2=\boxed{\mathrm{ (B)}\ 77}</math>. |
− | < | ||
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== See also == | == See also == | ||
− | {{AMC12 box|year=2002|ab=B|num-b=6|num- | + | {{AMC10 box|year=2002|ab=B|num-b=10|num-a=12}} |
+ | {{AMC12 box|year=2002|ab=B|num-b=6|num-a=8}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 23:01, 1 January 2020
- The following problem is from both the 2002 AMC 12B #7 and 2002 AMC 10B #11, so both problems redirect to this page.
Problem
The product of three consecutive positive integers is times their sum. What is the sum of their squares?
Solution
Let the three consecutive positive integers be , , and . Since the mean is , the sum of the integers is . So times the sum is just . With this, we now know that . , so . Hence, the sum of the squares is .
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.