If-1-1-3-1-3-5-1-5-7-1-7-9-1-2017-2019-x-y-where-x-y-is-in-its-lower-terms-find-the-value-of-x-y-

Question Number 126963 by ZiYangLee last updated on 25/Dec/20

If \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \frac{1}{7 \times 9} + \dots + \frac{1}{2017 \times 2019} = \frac{x}{y}

where \frac{x}{y} is in its lower terms,

find the value of x + y .

Answered by Olaf last updated on 25/Dec/20

S = \underset{n = 0}{\sum^{1008}} \frac{1}{(2 n + 1) (2 n + 3)}

S = \frac{1}{2} \underset{n = 0}{\sum^{1008}} [\frac{1}{2 n + 1} - \frac{1}{2 n + 3}]

S = \frac{1}{2} [1 - \frac{1}{2019}]

S = \frac{1}{2} \times \frac{2018}{2019} = \frac{1009}{2019}

x = 1009, y = 2019, x + y = 3028

Commented by ZiYangLee last updated on 30/Dec/20

perfect!