If for a G.P., a=5, r=2 and `S_(n)` =635 then : n =

Options

A. 5
B. 8
C. 7
D. none of these

Answer

To solve the problem, we need to find the value of \( n \) for a geometric progression (G.P.) where \( a = 5 \), \( r = 2 \), and \( S_n = 635 \).

### Step-by-step Solution:

1. **Write the formula for the sum of the first \( n \) terms of a G.P.**:
The formula for the sum of the first \( n \) terms \( S_n \) of a geometric progression is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

2. **Substitute the known values into the formula**:
Given \( a = 5 \), \( r = 2 \), and \( S_n = 635 \), we can substitute these values into the formula:
\[ 635 = \frac{5(2^n - 1)}{2 - 1} \]

3. **Simplify the equation**:
Since \( 2 - 1 = 1 \), the equation simplifies to:
\[ 635 = 5(2^n - 1) \]

4. **Divide both sides by 5**:
To isolate \( 2^n - 1 \), divide both sides by 5:
\[ \frac{635}{5} = 2^n - 1 \]
This gives:
\[ 127 = 2^n - 1 \]

5. **Add 1 to both sides**:
Now, add 1 to both sides to solve for \( 2^n \):
\[ 127 + 1 = 2^n \]
This results in:
\[ 128 = 2^n \]

6. **Express 128 as a power of 2**:
We know that \( 128 = 2^7 \). Therefore, we can equate the powers:
\[ 2^n = 2^7 \]

7. **Set the exponents equal to each other**:
Since the bases are the same, we can set the exponents equal:
\[ n = 7 \]

### Final Answer:
Thus, the value of \( n \) is \( 7 \).

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