If in the expansion of `(2^x+1/4^x)^n , T_3/T_2 = 7` and the sum of the co-efficients of `2nd` and `3rd` terms is `36,` then the value of `x` is

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. `-(1)/(4)`
D. `-(1)/(3)`

Answer

To solve the problem, we will follow the steps outlined in the video transcript and derive the value of \( x \) step by step.

### Step 1: Understanding the Terms in the Expansion
The expression given is \( (2^x + \frac{1}{4^x})^n \). We can rewrite \( \frac{1}{4^x} \) as \( \frac{1}{(2^2)^x} = \frac{1}{2^{2x}} \). Thus, the expression becomes:
\[ (2^x + 2^{-2x})^n \]

### Step 2: Finding the General Term
Using the Binomial Theorem, the general term \( T_r \) in the expansion can be expressed as:
\[ T_{r+1} = \binom{n}{r} (2^x)^{n-r} (2^{-2x})^r = \binom{n}{r} 2^{x(n-r)} 2^{-2xr} = \binom{n}{r} 2^{x(n - 3r)} \]

### Step 3: Finding \( T_2 \) and \( T_3 \)
- For \( T_2 \) (where \( r = 1 \)):
\[ T_2 = \binom{n}{1} 2^{x(n - 3 \cdot 1)} = n \cdot 2^{x(n - 3)} \]

- For \( T_3 \) (where \( r = 2 \)):
\[ T_3 = \binom{n}{2} 2^{x(n - 3 \cdot 2)} = \frac{n(n-1)}{2} \cdot 2^{x(n - 6)} \]

### Step 4: Setting Up the Ratio
We are given that \( \frac{T_3}{T_2} = 7 \):
\[ \frac{\frac{n(n-1)}{2} \cdot 2^{x(n - 6)}}{n \cdot 2^{x(n - 3)}} = 7 \]
This simplifies to:
\[ \frac{(n-1)}{2} \cdot 2^{-3x} = 7 \]
Multiplying both sides by 2 gives:
\[ (n-1) \cdot 2^{-3x} = 14 \]
Thus, we have:
\[ n - 1 = 14 \cdot 2^{3x} \quad \text{(Equation 1)} \]

### Step 5: Sum of Coefficients of \( T_2 \) and \( T_3 \)
The sum of the coefficients of \( T_2 \) and \( T_3 \) is given as 36:
\[ n + \frac{n(n-1)}{2} = 36 \]
Multiplying through by 2 to eliminate the fraction:
\[ 2n + n(n-1) = 72 \]
This simplifies to:
\[ n^2 + n - 72 = 0 \quad \text{(Equation 2)} \]

### Step 6: Solving the Quadratic Equation
Factoring Equation 2:
\[ (n + 9)(n - 8) = 0 \]
Thus, \( n = 8 \) (since \( n = -9 \) is not valid).

### Step 7: Substituting \( n \) Back
Substituting \( n = 8 \) into Equation 1:
\[ 8 - 1 = 14 \cdot 2^{3x} \]
This gives:
\[ 7 = 14 \cdot 2^{3x} \]
Dividing both sides by 14:
\[ \frac{1}{2} = 2^{3x} \]
This can be rewritten as:
\[ 2^{-1} = 2^{3x} \]
Thus, equating the exponents:
\[ -1 = 3x \]
So:
\[ x = -\frac{1}{3} \]

### Final Answer
The value of \( x \) is:
\[ \boxed{-\frac{1}{3}} \]