A G.P. of 64 terms is such that `S_n`(total)=`7(S_n)_("odd terms")` then common ratio of G.P is

Options

A. 2
B. 6
C. 10
D. 14

Answer

To solve the problem, we need to find the common ratio \( r \) of a geometric progression (G.P.) with 64 terms, given that the total sum \( S_n \) is equal to \( 7 \) times the sum of the odd terms.

### Step-by-Step Solution:

1. **Understanding the G.P.**:
A G.P. with first term \( a \) and common ratio \( r \) can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- ...
- 64th term: \( ar^{63} \)

2. **Finding the Total Sum \( S_{64} \)**:
The formula for the sum of the first \( n \) terms of a G.P. is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
For \( n = 64 \):
\[ S_{64} = \frac{a(r^{64} - 1)}{r - 1} \]

3. **Finding the Sum of Odd Terms**:
The odd terms in the G.P. are \( a, ar^2, ar^4, \ldots, ar^{62} \). This forms another G.P. with:
- First term: \( a \)
- Common ratio: \( r^2 \)
- Number of terms: 32 (since there are 64 terms total, half will be odd)

The sum of the odd terms \( S_{\text{odd}} \) is:
\[ S_{\text{odd}} = \frac{a( (r^2)^{32} - 1)}{r^2 - 1} = \frac{a(r^{64} - 1)}{r^2 - 1} \]

4. **Setting Up the Equation**:
According to the problem:
\[ S_{64} = 7 \cdot S_{\text{odd}} \]
Substituting the expressions for \( S_{64} \) and \( S_{\text{odd}} \):
\[ \frac{a(r^{64} - 1)}{r - 1} = 7 \cdot \frac{a(r^{64} - 1)}{r^2 - 1} \]
Since \( a \) and \( (r^{64} - 1) \) are common on both sides, we can cancel them out (assuming \( a \neq 0 \) and \( r^{64} \neq 1 \)):
\[ \frac{1}{r - 1} = \frac{7}{r^2 - 1} \]

5. **Cross Multiplying**:
Cross multiplying gives:
\[ r^2 - 1 = 7(r - 1) \]
Expanding this:
\[ r^2 - 1 = 7r - 7 \]
Rearranging gives:
\[ r^2 - 7r + 6 = 0 \]

6. **Factoring the Quadratic**:
Factoring the quadratic equation:
\[ (r - 6)(r - 1) = 0 \]
This gives us two possible solutions:
\[ r = 6 \quad \text{or} \quad r = 1 \]

7. **Conclusion**:
Since \( r = 1 \) would mean all terms are the same (not a typical G.P.), we take:
\[ r = 6 \]

### Final Answer:
The common ratio \( r \) of the G.P. is \( 6 \).