The odd value of n for which `704+1/2(704)`+… upto n terms = `1984-1/2(1984) +1/4(1984)` -… up to n terms is :

Options

A. 5
B. 3
C. 7
D. 9

Answer

To solve the equation given in the question, we need to analyze both sides of the equation step by step.

### Step 1: Write the series on the left-hand side
The left-hand side of the equation is:
\[ 704 + \frac{1}{2}(704) + \frac{1}{4}(704) + \ldots \text{ (up to n terms)} \]

This is a geometric series where:
- First term \( a = 704 \)
- Common ratio \( r = \frac{1}{2} \)

The sum of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Substituting the values:
\[ S_n = 704 \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = 704 \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = 704 \cdot 2 \left(1 - \left(\frac{1}{2}\right)^n\right) = 1408 \left(1 - \left(\frac{1}{2}\right)^n\right) \]

### Step 2: Write the series on the right-hand side
The right-hand side of the equation is:
\[ 1984 - \frac{1}{2}(1984) + \frac{1}{4}(1984) - \ldots \text{ (up to n terms)} \]

This series can be rewritten as:
\[ 1984 \left(1 - \frac{1}{2} + \frac{1}{4} - \ldots \right) \]

This is also a geometric series where:
- First term \( a = 1984 \)
- Common ratio \( r = -\frac{1}{2} \)

The sum of the first \( n \) terms of this series is:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Substituting the values:
\[ S_n = 1984 \frac{1 - \left(-\frac{1}{2}\right)^n}{1 - \left(-\frac{1}{2}\right)} = 1984 \frac{1 - \left(-\frac{1}{2}\right)^n}{\frac{3}{2}} = \frac{1984 \cdot 2}{3} \left(1 - \left(-\frac{1}{2}\right)^n\right) = \frac{3968}{3} \left(1 - \left(-\frac{1}{2}\right)^n\right) \]

### Step 3: Set the two sides equal to each other
Now we equate the two sums:
\[ 1408 \left(1 - \left(\frac{1}{2}\right)^n\right) = \frac{3968}{3} \left(1 - \left(-\frac{1}{2}\right)^n\right) \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 1408 \cdot 3 \left(1 - \left(\frac{1}{2}\right)^n\right) = 3968 \left(1 - \left(-\frac{1}{2}\right)^n\right) \]
\[ 4224 - 4224 \left(\frac{1}{2}\right)^n = 3968 - 3968 \left(-\frac{1}{2}\right)^n \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 4224 - 3968 = 4224 \left(\frac{1}{2}\right)^n - 3968 \left(-\frac{1}{2}\right)^n \]
\[ 256 = 4224 \left(\frac{1}{2}\right)^n + 3968 \left(\frac{1}{2}\right)^n \]
\[ 256 = (4224 + 3968) \left(\frac{1}{2}\right)^n \]
\[ 256 = 8192 \left(\frac{1}{2}\right)^n \]

### Step 6: Solve for \( n \)
Dividing both sides by 8192:
\[ \left(\frac{1}{2}\right)^n = \frac{256}{8192} \]
\[ \left(\frac{1}{2}\right)^n = \frac{1}{32} \]
Since \( \frac{1}{32} = \left(\frac{1}{2}\right)^5 \), we have:
\[ n = 5 \]

### Conclusion
The odd value of \( n \) for which the equation holds true is:
\[ \boxed{5} \]