Answer:
The next number of the series 0, 1/3, 1/2, 3/5, and 2/3 is 5/7
Step-by-step explanation:
The given numbers are;
0, 1/3, 1/2, 3/5, and 2/3
The number sequence is formed adding
to each (n - 1)th term to get the nth term number in the sequence, with the first term equal to 0, as follows;
For the 2nd term, the (n - 1)th term is 0, and n = 2, gives;
The
![0 +\dfrac{1}{\left (\dfrac{2^2 + 2}{2} \right ) } = 0 + \dfrac{1}{3} = \dfrac{1}{3}](https://tex.z-dn.net/?f=0%20%2B%5Cdfrac%7B1%7D%7B%5Cleft%20%28%5Cdfrac%7B2%5E2%20%2B%202%7D%7B2%7D%20%5Cright%20%29%20%7D%20%3D%200%20%2B%20%5Cdfrac%7B1%7D%7B3%7D%20%3D%20%5Cdfrac%7B1%7D%7B3%7D)
For the 3rd term, the (n - 1)th term is 1/3, and n = 3, gives;
![\dfrac{1}{3} +\dfrac{1}{\left (\dfrac{3^2 + 3}{2} \right ) } = \dfrac{1}{3} + \dfrac{1}{6} = \dfrac{1}{2}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B3%7D%20%2B%5Cdfrac%7B1%7D%7B%5Cleft%20%28%5Cdfrac%7B3%5E2%20%2B%203%7D%7B2%7D%20%5Cright%20%29%20%7D%20%3D%20%5Cdfrac%7B1%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B6%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D)
For the 4th term, the (n - 1)th term is 1/2, and n = 4, gives;
![\dfrac{1}{2} +\dfrac{1}{\left (\dfrac{4^2 + 4}{2} \right ) } = \dfrac{1}{2} + \dfrac{1}{10} = \dfrac{3}{5}](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7D%20%2B%5Cdfrac%7B1%7D%7B%5Cleft%20%28%5Cdfrac%7B4%5E2%20%2B%204%7D%7B2%7D%20%5Cright%20%29%20%7D%20%3D%20%5Cdfrac%7B1%7D%7B2%7D%20%2B%20%5Cdfrac%7B1%7D%7B10%7D%20%3D%20%5Cdfrac%7B3%7D%7B5%7D)
For the 5th term, the (n - 1)th term is 3/5, and n = 5, gives;
![\dfrac{3}{5} +\dfrac{1}{\left (\dfrac{5^2 + 5}{2} \right ) } = \dfrac{3}{5} + \dfrac{1}{15} = \dfrac{2}{3}](https://tex.z-dn.net/?f=%5Cdfrac%7B3%7D%7B5%7D%20%2B%5Cdfrac%7B1%7D%7B%5Cleft%20%28%5Cdfrac%7B5%5E2%20%2B%205%7D%7B2%7D%20%5Cright%20%29%20%7D%20%3D%20%5Cdfrac%7B3%7D%7B5%7D%20%2B%20%5Cdfrac%7B1%7D%7B15%7D%20%3D%20%5Cdfrac%7B2%7D%7B3%7D)
For the next or 6th term, the (n - 1)th term is 2/3, and n = 6, gives;
![\dfrac{2}{3} +\dfrac{1}{\left (\dfrac{6^2 + 6}{2} \right ) } = \dfrac{2}{3} + \dfrac{1}{21} = \dfrac{15}{21} = \dfrac{5}{7}](https://tex.z-dn.net/?f=%5Cdfrac%7B2%7D%7B3%7D%20%2B%5Cdfrac%7B1%7D%7B%5Cleft%20%28%5Cdfrac%7B6%5E2%20%2B%206%7D%7B2%7D%20%5Cright%20%29%20%7D%20%3D%20%5Cdfrac%7B2%7D%7B3%7D%20%2B%20%5Cdfrac%7B1%7D%7B21%7D%20%3D%20%20%5Cdfrac%7B15%7D%7B21%7D%20%3D%20%5Cdfrac%7B5%7D%7B7%7D)
The next number of the series 0, 1/3, 1/2, 3/5, and 2/3 = 5/7.