The numbers `3^(2 sin 2theta - 1), 14, 3^(4 - 2 sin 2 theta)` form first three terms of an A.P. Its fifth term is equal to

Options

A. -25
B. -12
C. 40
D. 53

Answer

To solve the problem, we need to find the fifth term of the arithmetic progression (A.P.) formed by the numbers \(3^{2 \sin 2\theta - 1}\), \(14\), and \(3^{4 - 2 \sin 2\theta}\).

### Step-by-Step Solution:

1. **Identify the terms of the A.P.:**
The first three terms are:
- \(a_1 = 3^{2 \sin 2\theta - 1}\)
- \(a_2 = 14\)
- \(a_3 = 3^{4 - 2 \sin 2\theta}\)

2. **Use the property of A.P.:**
For three numbers to be in A.P., the middle term must be equal to the average of the other two terms. This gives us the equation:
\[ 2a_2 = a_1 + a_3 \]
Substituting the values:
\[ 2 \cdot 14 = 3^{2 \sin 2\theta - 1} + 3^{4 - 2 \sin 2\theta} \]
This simplifies to:
\[ 28 = 3^{2 \sin 2\theta - 1} + 3^{4 - 2 \sin 2\theta} \]

3. **Let \(t = 3^{2 \sin 2\theta}\):**
Rewrite the equation in terms of \(t\):
\[ 28 = \frac{t}{3} + \frac{81}{t} \]
(since \(3^{4 - 2 \sin 2\theta} = \frac{3^4}{3^{2 \sin 2\theta}} = \frac{81}{t}\))

4. **Multiply through by \(3t\) to eliminate the fractions:**
\[ 28 \cdot 3t = t^2 + 243 \]
This simplifies to:
\[ 84t = t^2 + 243 \]
Rearranging gives:
\[ t^2 - 84t + 243 = 0 \]

5. **Solve the quadratic equation:**
Using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ t = \frac{84 \pm \sqrt{84^2 - 4 \cdot 1 \cdot 243}}{2 \cdot 1} \]
Calculate the discriminant:
\[ 84^2 = 7056, \quad 4 \cdot 243 = 972 \quad \Rightarrow \quad 7056 - 972 = 6084 \]
Now calculate \(t\):
\[ t = \frac{84 \pm \sqrt{6084}}{2} \]
Since \(\sqrt{6084} = 78\):
\[ t = \frac{84 \pm 78}{2} \]
This gives two values:
\[ t = \frac{162}{2} = 81 \quad \text{and} \quad t = \frac{6}{2} = 3 \]

6. **Choose the valid value for \(t\):**
We reject \(t = 81\) because it leads to an invalid sine value. Thus, we take \(t = 3\):
\[ 3^{2 \sin 2\theta} = 3 \quad \Rightarrow \quad 2 \sin 2\theta = 1 \quad \Rightarrow \quad \sin 2\theta = \frac{1}{2} \]

7. **Find the first term and common difference:**
Substitute \(t = 3\) back to find the first term:
\[ a_1 = 3^{2 \sin 2\theta - 1} = 3^{0} = 1 \]
The third term:
\[ a_3 = 3^{4 - 2 \sin 2\theta} = 3^{4 - 1} = 3^{3} = 27 \]
The common difference \(d\) can be found as:
\[ d = a_2 - a_1 = 14 - 1 = 13 \]

8. **Calculate the fifth term:**
The fifth term \(a_5\) of the A.P. is given by:
\[ a_5 = a_1 + 4d = 1 + 4 \cdot 13 = 1 + 52 = 53 \]

### Final Answer:
The fifth term is \(53\).