Consider the expansion `( x^(2) + ( 1)/( x))^(15)`
The sum of the coefficients of all the terms in the expansion is

Options

A. `2^(2n-1)`
B. `4^(n-1)`
C. `2 xx 4^(n)`
D. None of these

Answer

To find the sum of the coefficients of all the terms in the expansion of \( (x^2 + \frac{1}{x})^{15} \), we can use a shortcut method based on the Binomial Theorem. Here’s a step-by-step solution:

### Step 1: Identify the expression
We start with the expression:
\[ (x^2 + \frac{1}{x})^{15} \]

### Step 2: Recognize the coefficients
In this expression, we can identify \( a = 1 \) (coefficient of \( x^2 \)), \( b = 1 \) (coefficient of \( \frac{1}{x} \)), and \( c = 0 \) (since there is no constant term).

### Step 3: Use the formula for the sum of coefficients
To find the sum of the coefficients of the expansion, we can use the formula:
\[ \text{Sum of coefficients} = (a + b + c)^n \]
where \( n \) is the exponent of the binomial expression.

### Step 4: Substitute the values
Substituting the values we identified:
\[ a = 1, \quad b = 1, \quad c = 0, \quad n = 15 \]
we get:
\[ \text{Sum of coefficients} = (1 + 1 + 0)^{15} = (2)^{15} \]

### Step 5: Calculate \( 2^{15} \)
Now, we calculate \( 2^{15} \):
\[ 2^{15} = 32768 \]

### Conclusion
Thus, the sum of the coefficients of all the terms in the expansion \( (x^2 + \frac{1}{x})^{15} \) is:
\[ \boxed{32768} \]