Find the ratio of the coefficient of `x^(15)` to the term independent of x in the expansion of `(x^(2)+2/x)^(15)`.

Options

A. `1:4`
B. `1:32`
C. `7:64`
D. `7:16`

Answer

To solve the problem of finding the ratio of the coefficient of \( x^{15} \) to the term independent of \( x \) in the expansion of \( (x^2 + \frac{2}{x})^{15} \), we will follow these steps:

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the expansion of \( (x^2 + \frac{2}{x})^{15} \) can be expressed as:
\[ T_{r+1} = \binom{15}{r} (x^2)^{15-r} \left(\frac{2}{x}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{15}{r} (x^2)^{15-r} \cdot \frac{2^r}{x^r} = \binom{15}{r} 2^r x^{30 - 3r} \]

### Step 2: Find the Coefficient of \( x^{15} \)
To find the coefficient of \( x^{15} \), we set the exponent of \( x \) equal to 15:
\[ 30 - 3r = 15 \]
Solving for \( r \):
\[ 3r = 30 - 15 \implies 3r = 15 \implies r = 5 \]
Now, substituting \( r = 5 \) into the general term to find the coefficient:
\[ T_{6} = \binom{15}{5} 2^5 x^{15} \]
Thus, the coefficient of \( x^{15} \) is:
\[ \text{Coefficient of } x^{15} = \binom{15}{5} 2^5 \]

### Step 3: Find the Term Independent of \( x \)
Next, we need to find the term independent of \( x \). For this, we set the exponent of \( x \) to 0:
\[ 30 - 3r = 0 \]
Solving for \( r \):
\[ 3r = 30 \implies r = 10 \]
Now, substituting \( r = 10 \) into the general term to find the coefficient:
\[ T_{11} = \binom{15}{10} 2^{10} x^{0} \]
Thus, the term independent of \( x \) has a coefficient of:
\[ \text{Coefficient of the term independent of } x = \binom{15}{10} 2^{10} \]

### Step 4: Calculate the Ratio
Now we can find the ratio of the coefficient of \( x^{15} \) to the term independent of \( x \):
\[ \text{Ratio} = \frac{\binom{15}{5} 2^5}{\binom{15}{10} 2^{10}} \]
Using the property \( \binom{n}{r} = \binom{n}{n-r} \):
\[ \binom{15}{10} = \binom{15}{5} \]
Thus, the ratio simplifies to:
\[ \text{Ratio} = \frac{2^5}{2^{10}} = \frac{1}{2^{10-5}} = \frac{1}{2^5} = \frac{1}{32} \]

### Final Answer
The ratio of the coefficient of \( x^{15} \) to the term independent of \( x \) is:
\[ \text{Ratio} = 1 : 32 \]