The term independent of x in the expansion of
`(1-1/(2x)+3x^(7))(x^(2)+1/(3x))^(10)` is

Options

A. `10//3^(8)`
B. `-2//3`
C. `10//3^(9)`
D. `11//6`

Answer

To find the term independent of \( x \) in the expansion of

\[ (1 - \frac{1}{2x} + 3x^7) \left( x^2 + \frac{1}{3x} \right)^{10}, \]

we will follow these steps:

### Step 1: Expand \( \left( x^2 + \frac{1}{3x} \right)^{10} \)

The general term in the binomial expansion of \( (a + b)^n \) is given by:

\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r. \]

Here, \( a = x^2 \), \( b = \frac{1}{3x} \), and \( n = 10 \). Therefore, the general term is:

\[ T_{r+1} = \binom{10}{r} (x^2)^{10-r} \left( \frac{1}{3x} \right)^r = \binom{10}{r} \frac{1}{3^r} x^{20 - 3r}. \]

### Step 2: Identify the term independent of \( x \)

We need to find \( r \) such that the power of \( x \) is zero:

\[ 20 - 3r = 0 \implies 3r = 20 \implies r = \frac{20}{3}. \]

Since \( r \) must be an integer, we check the next possible integer values.

### Step 3: Check for integer values of \( r \)

The next integer values to check are \( r = 6 \) and \( r = 7 \):

1. For \( r = 6 \):
\[ 20 - 3(6) = 20 - 18 = 2 \quad (\text{not independent}) \]

2. For \( r = 7 \):
\[ 20 - 3(7) = 20 - 21 = -1 \quad (\text{not independent}) \]

### Step 4: Consider the term \( 3x^7 \)

Next, we consider the term \( 3x^7 \) from \( (1 - \frac{1}{2x} + 3x^7) \):

When we multiply \( 3x^7 \) with the term from the binomial expansion, we need to find \( r \) such that:

\[ 20 - 3r + 7 = 0 \implies 27 - 3r = 0 \implies 3r = 27 \implies r = 9. \]

### Step 5: Calculate the term for \( r = 9 \)

Substituting \( r = 9 \) into the general term:

\[ T_{10} = \binom{10}{9} \frac{1}{3^9} x^{20 - 27} = \binom{10}{9} \frac{1}{3^9} x^{-7}. \]

Now, we need to combine this with \( 3x^7 \):

\[ 3x^7 \cdot T_{10} = 3 \cdot \binom{10}{9} \cdot \frac{1}{3^9} x^{0} = 3 \cdot 10 \cdot \frac{1}{3^9}. \]

### Step 6: Final Calculation

Thus, the term independent of \( x \) is:

\[ \frac{30}{3^9} = \frac{30}{19683}. \]

### Conclusion

The term independent of \( x \) in the expansion is:

\[ \frac{30}{19683}. \]