Solution:

We need to find out the number of terms of the given AP that must be taken to give a sum of 636.

Let the number of terms be n.

The formula to find the sum of first n terms of an AP is given by:

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

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

Substituting the given values, we get:

636 = n/2 [2(9) + (n-1)8]

Simplifying the above equation, we get:

1272 = n[18 + 8(n-1)]

1272 = 8n^2 + 10n - 18

8n^2 + 10n - 1290 = 0

Dividing both sides by 2, we get:

4n^2 + 5n - 645 = 0

Solving the above quadratic equation using the quadratic formula, we get:

n = 15 or n = -10.125

Since the number of terms cannot be negative, we reject n = -10.125.

Hence, the number of terms of the given AP that must be taken to give a sum of 636 is 15.

Answer: The number of terms of the AP: 9, 17, 25 that must be taken to give a sum of 636 is 15.