Let us rewrtie the number 5.313131... as a series:

5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + ....

We notice that the terms:

0.31 + 0.0031 + 0.000031 ... are terms of a geometric series where a1 = 0.13 and common difference r = 0.01.

Then , we know tha the sum of the G.S is given as follows:

S = a1/ (1- r)

Let us substitute:

= 0.31/ ( 1- 0.01)

= 0.31 / 0.99

= 31/99

Then we conclude that we could rewrite the number as follows:

5.313131... = 5 + 31/ 99

= ( 495 + 31)/ 99 = 526/ 99

**==> 5. 313131... = 526/ 99**

The given rational number 5.313131....

Let S = 5.31313131.....(1)

We mutiply both sides by 100 and we get:

100S = 531.313131.....(2).

We subtract (1) from (2) and we get:

100S - S = (531.31313131....) - (5.31313131....)

99S = 526.

S= 526/99

Therefore the give number 5.313131.... = 526/99 which is in the form of p/q where both pand q are rational numbers.