To solve this equation, we can use the formula for the sum of an arithmetic series:

S = (n/2)(a + l)

where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

In this case, the first term (a) is x.3 and the last term (l) is x.99. The common difference between terms is 0.2 (0.5 - 0.3 = 0.2).

We need to find the value of x that makes the sum equal to 7497. So we can set up the equation:

7497 = (n/2)(x.3 + x.99)

Simplifying further:

7497 = (n/2)(1.29x)

Now, let's solve for n:

n = (7497 * 2) / (1.29x)

n = 14994 / (1.29x)

Since n must be a positive integer, we can try different values of x to find a solution.

Let's start with x = 1:

n = 14994 / (1.29 * 1)
n = 11627.91

Since n must be a positive integer, x = 1 is not a solution.

Let's try x = 2:

n = 14994 / (1.29 * 2)
n = 5796.51

Again, n is not a positive integer.

We can continue trying different values of x until we find a solution where n is a positive integer.