**Problem Statement:**
Given an arithmetic progression (A.P) with an odd number of terms, the sum of all the terms is 15/8 times the sum of the terms in odd places. We need to find the number of terms in the A.P.

**Solution:**

Let's assume that the arithmetic progression has 'n' terms, where 'n' is an odd number.

**Step 1: Identify the variables**
Let's denote the first term of the A.P as 'a' and the common difference as 'd'. We need to find the value of 'n', the number of terms in the A.P.

**Step 2: Understand the sum of terms in an A.P**
The sum of 'n' terms in an A.P can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)

**Step 3: Express the given condition mathematically**
According to the given condition, the sum of all the terms is 15/8 times the sum of the terms in odd places. Mathematically, this can be expressed as:
Sn = (15/8)Sodd

**Step 4: Calculate the sum of terms in odd places**
The sum of terms in odd places can be calculated by considering only the odd terms of the A.P. Since 'n' is odd, the number of odd terms will also be 'n'. So, the sum of terms in odd places can be calculated using the formula:
Sodd = (n/2)(2a + (n-1)d)

**Step 5: Substitute the values and simplify**
Substituting the expressions for Sn and Sodd in the equation Sn = (15/8)Sodd, we get:
(n/2)(2a + (n-1)d) = (15/8)(n/2)(2a + (n-1)d)

Simplifying the equation, we get:
(2a + (n-1)d) = (15/8)(a + (n-1)d)

**Step 6: Solve for 'n'**
Let's solve the equation to find the value of 'n'.

2a + nd - d = (15/8)(a + nd - d)

Multiplying both sides of the equation by 8, we get:
16a + 8nd - 8d = 15a + 15nd - 15d

Rearranging the equation, we get:
15nd - 8nd = 15d - 16a + 8d - 15a

Simplifying the equation, we get:
7nd = 7d - a

Dividing both sides of the equation by 7, we get:
nd = d - (a/7)

Since 'n' and 'd' are integers, we can conclude that 'a' must be divisible by 7.

Hence, the number of terms in the A.P is 'n' which can be any odd number such that 'a' is divisible by 7.