SOLUTION: the sum of the reciprocals of 2 consecutive integers is 25 over 156. find the integers.

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Question 894823: the sum of the reciprocals of 2 consecutive integers is 25 over 156. find the integers.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
n and n+1 are the consecutive integers.
highlight_green%281%2Fn%2B1%2F%28n%2B1%29=25%2F156%29

Lowest common denominator is 156n(n+1); so multiply both sides by this.
156%28n%2B1%29%2B156n=25n%28n%2B1%29
156n%2B156%2B156n=25n%5E2%2B25n
312n%2B156=25n%5E2%2B25n
25n%5E2%2B25n=312n%2B156
2n%5E2=287n%2B156---THE MISTAKE HERE!!!
highlight_green%2825n%5E2-287n-156=0%29---should be factorable, but not worth the effort.

n=%28287%2B-+sqrt%28287%5E2%2B4%2A25%2A156%29%29%2F%282%2A25%29
n=%28287%2B-+sqrt%2897969%29%29%2F%282%2A25%29
NO. Not sensible. 83617 is not a square number. n will not be an integer.

Solution that works: The numbers are 12 and 13.