To solve this problem, we can use the formula for the sum of an arithmetic series.

The sum of an arithmetic series is given by the formula:

S = (n/2)(2a + (n-1)d)

where S is the sum of the series, n is the number of terms, a is the first term, and d is the common difference.

In this case, the first term is x+2, the common difference is 2, and the sum is 99800. We need to find the number of terms.

Using the formula, we can write the equation:

99800 = (n/2)(2(x+2) + (n-1)2)

Simplifying the equation, we get:

99800 = n(x+2) + n(n-1)

Expanding and rearranging the equation, we get:

99800 = nx + 2n + n^2 - n

Combining like terms, we get:

n^2 + (x+1)n + 2n - 99800 = 0

Simplifying further, we get:

n^2 + (x+3)n - 99800 = 0

Now, we can use the quadratic formula to solve for n:

n = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = (x+3), and c = -99800.

Plugging in the values, we get:

n = (-(x+3) ± √((x+3)^2 - 4(1)(-99800))) / (2(1))

Simplifying further, we get:

n = (-(x+3) ± √(x^2 + 6x + 9 + 399200)) / 2

n = (-(x+3) ± √(x^2 + 6x + 399209)) / 2

Now, we need to find the value of x that satisfies the equation. Since we are looking for a positive number of terms, we can ignore the negative solution.

Setting n = 1996, we can solve for x:

1996 = (-(x+3) + √(x^2 + 6x + 399209)) / 2

Multiplying both sides by 2, we get:

3992 = -(x+3) + √(x^2 + 6x + 399209)

Simplifying further, we get:

3992 + x + 3 = √(x^2 + 6x + 399209)

Squaring both sides, we get:

(3992 + x + 3)^2 = x^2 + 6x + 399209

Expanding and simplifying, we get:

15936649 + 7988x + 6x^2 + 11976x + x^2 = x^2 + 6x + 399209

Combining like terms, we get:

7x^2 + 19970x - 383560 = 0

Now, we can use the quadratic formula to solve for x:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 7, b = 19970, and c = -383560.

Plugging in the values, we get:

x = (-(19970) ± √((19970)^2 - 4(7)(-383560))) / (2(7))

Simplifying further, we get:

x = (-19970 ± √(398800900 - (-383560)(4)(7))) / 14

x = (-19970 ± √(398800900 + 10736960)) / 14

x = (-19970 ± √(409537860)) / 14

x = (-19970 ± 20236.82) / 14

Now, we can solve for x:

x = (-19970 + 20236.82) / 14

x = 266.82 / 14

x ≈ 19.057

Therefore, the value of x that satisfies the equation is approximately 19.057.