Question

What's the next number? 0 , 1/3 , 1/2 , 3/5 , 2/3​

Answer 1

The next number of the series 0, 1/3, 1/2, 3/5, and 2/3 is 5/7

Explanation:

The given numbers are;

0, 1/3, 1/2, 3/5, and 2/3

The number sequence is formed adding
`(1)/(\left ((n^2 + n)/(2) \right ) )` 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 +(1)/(\left ((2^2 + 2)/(2) \right ) ) = 0 + (1)/(3) = (1)/(3)`

For the 3rd term, the (n - 1)th term is 1/3, and n = 3, gives;

`(1)/(3) +(1)/(\left ((3^2 + 3)/(2) \right ) ) = (1)/(3) + (1)/(6) = (1)/(2)`

For the 4th term, the (n - 1)th term is 1/2, and n = 4, gives;

`(1)/(2) +(1)/(\left ((4^2 + 4)/(2) \right ) ) = (1)/(2) + (1)/(10) = (3)/(5)`

For the 5th term, the (n - 1)th term is 3/5, and n = 5, gives;

`(3)/(5) +(1)/(\left ((5^2 + 5)/(2) \right ) ) = (3)/(5) + (1)/(15) = (2)/(3)`

For the next or 6th term, the (n - 1)th term is 2/3, and n = 6, gives;

`(2)/(3) +(1)/(\left ((6^2 + 6)/(2) \right ) ) = (2)/(3) + (1)/(21) = (15)/(21) = (5)/(7)`

The next number of the series 0, 1/3, 1/2, 3/5, and 2/3 = 5/7.