SOLUTION: Find the sum of the series: 18/12 + 18/20 + 18/30 + 18/42 + ... + 18/1260.

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Question 1197721: Find the sum of the series: 18/12 + 18/20 + 18/30 + 18/42 + ... + 18/1260.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the sum of the series: 18/12 + 18/20 + 18/30 + 18/42 + ... + 18/1260.
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                                Step by step 


(1)  18/12 + 18/20 + 18/30 + 18/42 + ... + 18/1260 = 18*(1/12 + 1/20 + 1/30 + 1/42 + . . . + 1/1260).



(2)  Every addend in the parentheses is 

     1%2F12 = 1%2F%283%2A4%29 = 1%2F3 - 1%2F4

     1%2F20 = 1%2F%284%2A5%29 = 1%2F4 - 1%2F5

     1%2F30 = 1%2F%285%2A6%29 = 1%2F5 - 1%2F6

     1%2F42 = 1%2F%286%2A7%29 = 1%2F6 - 1%2F7

          . . . . . . . . . . . . . . . . 

     1%2F1260 = 1%2F%2835%2A36%29 = 1%2F35 - 1%2F36



(3)  Add fractions in n.2, on the left sides and on the right sides.

     On the left side, you will get the sum, which is in parentheses in n.1.

     On the right side, all interior term will cancel each other, and only two extreme terms will survive


        1/12 + 1/20 + 1/30 + 1/42 + . . . + 1/1260 = 1%2F3 - 1%2F36 = 12%2F36+-+1%2F36 = 11%2F36.



(4)  THEREFORE

        18/12 + 18/20 + 18/30 + 18/42 + ... + 18/1260 = 18%2A%2811%2F36%29 = 11%2F2.


ANSWER.  The requested sum is equal to 11%2F2 = 51%2F2 = 5.5.    ANSWER

Solved.

It is a Sunday's school Math circle problem.




Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

18/12 + 18/20 + 18/30 + ... + 18/1260

18(1/12 + 1/20 + 1/30 + ... + 1/1260)

18(1/(3*4) + 1/(4*5) + 1/(5*6) + ... + 1/(35*36)

18*sum%281%2F%28i%28i%2B1%29%29%2C3%2C35%29

18*sum%28%281%2Fi-1%2F%28i%2B1%29%29%2C3%2C35%29

18*((1/3-1/4) + (1/4-1/5) + (1/5-1/6) + ... + (1/34-1/35) + (1/35-1/36))

18(1/3-1/36) = 6-1/2 = 5.5

ANSWER: 5.5