1/3
3/6
5/9
7/12
9/15

After that

11/18
13/21

And so on

Method 1 :

Difference between each pair of consecutive fractions always has the numerator 1.

The denominator has this pattern :

½ - ⅓ = 3/6 - 2/6 = 1/6 = 1/(6x1)

5/9 - ½ = 10/18 - 9/18 = 1/18 = 1/(6x3) = 1/(6 x (1+2))

7/12 - 5/9 = 21/36 - 20/36 = 1/36 = 1/(6x6) = 1/(6x(1+2+3))

3/5 - 7/12 = 36/60 - 35/60 = 1/60 = 1/(6x10) = 1/(6x(1+2+3+4))

So to go to the next fraction,

3/5 + 1/(6x(1+2+3+4+5))

= 3/5 + 1/(6x15)

= 3/5 + 1/90

= 54/90 + 1/90

= 55/90

= 11/18

Method 2 :


Numerators keep increasing by 2. Denominators keep increasing by 3.


1/3

1/2 = 3/6 = (1+2)/(3+3)

5/9 = (3+2)/(6+3)

7/12 = (5+2)/(9+3)

3/5 = 9/15 = (7+2)/(12+3)

The next one will be : (9+2)/(15+3) = 11/18



The other way to see it is : the difference in numerator and denominator keeps increasing by 1 .


1/3 → 3 - 1 = 2

1/2 = 3/6 →6 - 3 = 3

5/9 → 9 - 5 = 4

7/12 → 12 - 7 = 5

3/5 = 9/15 →15 - 9 = 6

So 11/18 → 18 - 11 = 7