Question

1/1×3 + 1/3×5 + ... + 1/47×49 HELP PLZ

Answer 1

24/49

Explanation:

Let's add the terms and see if there's a pattern

`(1)/(1* 3)+(1)/(3* 5)=(5+1)/(1* 3* 5)=(2)/(5)\quad\text{sum of 2 terms}\\\\(2)/(5)+(1)/(5* 7)=(14+1)/(5*7)=(3)/(7)\quad\text{sum of 3 terms}`

Suppose we say the sum of n terms is (n/(2n+1)), the next term in the series will be 1/((2n+1)(2n+3)) and adding that to the presumed sum gives ...

`(n)/(2n+1)+(1)/((2n+1)(2n+3))=(n(2n+3)+1)/((2n+1)(2n+3))=(2n^2+3n+1)/((2n+1)(2n+3))\\\\=((2n+1)(n+1))/((2n+1)(2n+3))=(n+1)/(2n+3)\text{ matches }((n+1))/(2(n+1)+1)`

Then it appears the sum of n terms is (n/(2n+1)). So, the sum of 24 terms is ...

`S_(24)=(24)/(2*24+1)=\boxed{(24)/(49)}`