**Solution:**

Let's assume that the first term of the arithmetic progression (AP) is 'a' and the common difference is 'd'.

Given that the sum of a certain number of terms of the AP is 52.

To find the number of terms, we need to use the formula for the sum of the first 'n' terms of an AP:

**Sn = (n/2)(2a + (n - 1)d)**

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

In this case, the first term 'a' is -8 and the common difference 'd' is 2 (since each term is increasing by 2).

So, we have:

**52 = (n/2)(2(-8) + (n - 1)(2))**

Simplifying the equation further:

**52 = (n/2)(-16 + 2n - 2)**

**52 = (n/2)(2n - 18)**

**52 = (n/2)(2(n - 9))**

Dividing both sides of the equation by 2:

**26 = n(n - 9)**

This is a quadratic equation. Rearranging it:

**n^2 - 9n - 26 = 0**

To solve this quadratic equation, we can factorize or use the quadratic formula. Let's use the quadratic formula:

**n = (-b ± √(b^2 - 4ac))/(2a)**

For our equation, a = 1, b = -9, and c = -26.

**n = (-(-9) ± √((-9)^2 - 4(1)(-26)))/(2(1))**

Simplifying further:

**n = (9 ± √(81 + 104))/(2)**

**n = (9 ± √(185))/(2)**

The discriminant (b^2 - 4ac) is 185, which is a positive number. So, the quadratic equation has two real and distinct roots.

Using the quadratic formula, we have two possible values for 'n':

**n = (9 + √(185))/(2)** or **n = (9 - √(185))/(2)**

Calculating these values using a calculator or approximating:

**n ≈ 10.94** or **n ≈ -1.94**

Since the number of terms cannot be negative, the number of terms is approximately 10.

Therefore, the number of terms in the arithmetic progression is 10.

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