Question

The term independent of $ x $ in the expansion of $ \left( x - \frac{3}{x^2} \right)^{18} $

• A. \( C_6^{18} \)
• B. \( C_6 \cdot 3^6 \)
• C. \( C_6 \cdot 3^{-6} \)
• D. \( 3^6 \)

Hint:

In binomial expansions, the term independent of \( x \) corresponds to the value of \( r \) that makes the exponent of \( x \) zero.

Answer 1

The Correct Option is C

Step 1: Expanding the Binomial Expression
We use the binomial expansion for \( (a + b)^n \): \[ \left( x - \frac{3}{x^2} \right)^{18} = \sum_{r=0}^{18} C_{18}^{r} x^{18-r} \left(-\frac{3}{x^2}\right)^r \] Simplifying the powers of \( x \): \[ = \sum_{r=0}^{18} C_{18}^{r} (-3)^r x^{18 - r - 2r} = \sum_{r=0}^{18} C_{18}^{r} (-3)^r x^{18 - 3r} \]
Step 2: Finding the Term Independent of \( x \)
The term independent of \( x \) will occur when the exponent of \( x \) is zero, i.e., when \( 18 - 3r = 0 \).
Solving for \( r \): \[ r = 6 \] Substituting \( r = 6 \) into the binomial expansion: \[ C_{18}^{6} (-3)^6 = C_6 \cdot 3^6 \]
Step 3: Conclusion
Thus, the term independent of \( x \) is \( C_6 \cdot 3^{-6} \).