2016 AMC 8 Problems/Problem 19
Contents
Problem
The sum of consecutive even integers is . What is the largest of these consecutive integers?
Solution 1
Let be the 13th consecutive even integer that's being added up. By doing this, we can see that the sum of all 25 even numbers will simplify to since . Now, . Remembering that this is the 13th integer, we wish to find the 25th, which is .
Solution 2
Let be the largest number. Then, . Factoring this gives . Grouping like terms gives , and continuing down the line, we find .
~MrThinker
Solution 3
Let be the smallest number. The equation will become, . After you combine like terms, you get which turns into . , so . Then, you add .
Note: After we combine like terms, you would have an arithmetic sequence from to (because to get last term), which would look like To calculate the sum of the numbers we can use the formula . This simplifies to , giving us , which is what AfterglowBlaziken did.
~AfterglowBlaziken ~ Note by probab2023
Solution 4
Dividing the series by , we get that the sum of consecutive integers is . Let the middle number be we know that the sum is , so . Solving, . is the middle term of the original sequence, so the original last term is . So the answer is .
~vadava_lx
Video Solution (CREATIVE THINKING + ANALYSIS!!!)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2016 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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All AJHSME/AMC 8 Problems and Solutions |
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