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Compute \(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} + \dots + \frac{1}{97 \times 100}\)

 May 5, 2023
 #1
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By telescoping sums, this is equal to 1/3.

 May 6, 2023
 #2
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a=listfor(n, 0, 32 ,1 / ((1+3*n)*(1+3*(n+1)));print a, "==",sum a

 

(1 / 4, 1 / 28, 1 / 70, 1 / 130, 1 / 208, 1 / 304, 1 / 418, 1 / 550, 1 / 700, 1 / 868, 1 / 1054, 1 / 1258, 1 / 1480, 1 / 1720, 1 / 1978, 1 / 2254, 1 / 2548, 1 / 2860, 1 / 3190, 1 / 3538, 1 / 3904, 1 / 4288, 1 / 4690, 1 / 5110, 1 / 5548, 1 / 6004, 1 / 6478, 1 / 6970, 1 / 7480, 1 / 8008, 1 / 8554, 1 / 9118, 1 / 9700) == 33 / 100

 May 6, 2023

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