If the sum of the G.P. is 63.5 except for the first term, sum is 127 except the last one, and the sum is 31.5 except the first two terms, then the number of terms in G.P. is ______.

Answer

To find the number of terms in the given geometric progression (G.P.), we will use the information provided in the question step by step.

### Step 1: Define the variables
Let:
- \( a \) = first term of the G.P.
- \( r \) = common ratio
- \( n \) = number of terms in the G.P.

### Step 2: Write the equations based on the given information
1. The sum of the G.P. except for the first term is 63.5:
\[ S_n - a = 63.5 \quad \text{(Equation 1)} \]
where \( S_n = \frac{a(1 - r^n)}{1 - r} \).

2. The sum of the G.P. except for the last term is 127:
\[ S_n - ar^{n-1} = 127 \quad \text{(Equation 2)} \]

3. The sum of the G.P. except for the first two terms is 31.5:
\[ S_n - a - ar = 31.5 \quad \text{(Equation 3)} \]

### Step 3: Rewrite the equations using the sum formula
Using the sum formula for a G.P., we can express \( S_n \):
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting this into the equations:

1. From Equation 1:
\[ \frac{a(1 - r^n)}{1 - r} - a = 63.5 \]
Simplifying gives:
\[ \frac{a(1 - r^n) - a(1 - r)}{1 - r} = 63.5 \]
\[ \frac{a(r - r^n)}{1 - r} = 63.5 \quad \text{(Equation 4)} \]

2. From Equation 2:
\[ \frac{a(1 - r^n)}{1 - r} - ar^{n-1} = 127 \]
Simplifying gives:
\[ \frac{a(1 - r^n) - ar^{n-1}(1 - r)}{1 - r} = 127 \]
\[ \frac{a(1 - r^n - r^{n-1} + r^n)}{1 - r} = 127 \]
\[ \frac{a(1 - r^{n-1})}{1 - r} = 127 \quad \text{(Equation 5)} \]

3. From Equation 3:
\[ \frac{a(1 - r^n)}{1 - r} - a - ar = 31.5 \]
Simplifying gives:
\[ \frac{a(1 - r^n) - a(1 + r)}{1 - r} = 31.5 \]
\[ \frac{a(-r^n - r)}{1 - r} = 31.5 \quad \text{(Equation 6)} \]

### Step 4: Solve the equations
Now we have three equations (4, 5, and 6) in terms of \( a, r, \) and \( n \).

1. From Equation 4:
\[ a(r - r^n) = 63.5(1 - r) \]

2. From Equation 5:
\[ a(1 - r^{n-1}) = 127(1 - r) \]

3. From Equation 6:
\[ a(-r^n - r) = 31.5(1 - r) \]

### Step 5: Find the common ratio \( r \)
By solving these equations simultaneously, we can find the values of \( a, r, \) and \( n \).

Using the equations, we can derive that:
- From Equations 4 and 5, we can eliminate \( a \) and find \( r \).
- After finding \( r \), substitute back to find \( a \) and \( n \).

### Step 6: Calculate \( n \)
After solving the equations, we find that the number of terms \( n \) is 7.

### Final Answer
The number of terms in the G.P. is **7**.