The given series is an arithmetic progression with a common difference of 2. The first term of the series is -8 and the nth term is -2n-10.


The formula to find the sum of n terms of an arithmetic progression is:

S_n = n/2 [2a + (n-1)d]

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


Let the number of terms be n.

The first term of the series is -8 and the common difference is 2.

Therefore, the nth term of the series is -2n-10.

Using the formula, we can find the sum of the series:

S_n = n/2 [2a + (n-1)d]

S_n = n/2 [2(-8) + (n-1)2]

S_n = n/2 [-16 + 2n - 2]

S_n = n/2 [2n - 18]

Simplifying the expression:

2S_n = n [2n - 18]

2S_n = 2n² - 18n

S_n = n² - 9n

Given that the sum of the series is 52:

n² - 9n = 52

n² - 9n - 52 = 0

Solving the quadratic equation, we get:

n = 13 or n = -4

Since the number of terms cannot be negative, the number of terms in the series is 13.

Therefore, the number of terms n is 13.