https://file.aiquickdraw.com/mathbot/file/1726006833740b15cb6ba.jpg

The image asks for the "third term of \( \left(a + \frac{4}{a^2}\right)^{11} \)" using binomial expansion. Let's break it down and find the third term.

### Step-by-Step Approach:
1. **Binomial Expansion Formula**: 
   The binomial expansion of \( (x + y)^n \) is given by:

   \[
   (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
   \]

   For this specific case:
   - \( x = a \)
   - \( y = \frac{4}{a^2} \)
   - \( n = 11 \)

2. **Finding the Third Term**:
   The third term corresponds to \( k = 2 \) (since terms are indexed from \( k=0 \)):

   \[
   \text{Third term} = \binom{11}{2} a^{11-2} \left(\frac{4}{a^2}\right)^2
   \]

3. **Simplify**:
   - \( \binom{11}{2} = \frac{11 \times 10}{2} = 55 \)
   - \( a^{11-2} = a^9 \)
   - \( \left(\frac{4}{a^2}\right)^2 = \frac{16}{a^4} \)

   Therefore, the third term is:

   \[
   55 \cdot a^9 \cdot \frac{16}{a^4} = 55 \cdot 16 \cdot a^5 = 880 a^5
   \]

### Conclusion:
The third term of the expansion of \( \left(a + \frac{4}{a^2}\right)^{11} \) is \( 880a^5 \).

---

Would you like further details or have any questions about this process?

Here are some related questions:
1. How do you compute binomial coefficients like \( \binom{11}{2} \)?
2. What is the general form of binomial expansion for negative or fractional powers?
3. How would the fourth term of this binomial expansion be calculated?
4. Why does the index \( k=2 \) correspond to the third term?
5. How does the power of \( a \) change for different terms in the binomial expansion?

**Tip**: Always keep track of powers carefully when using the binomial expansion, especially when dealing with fractions.

Learn how to find the third term of the binomial expansion of (a + 4/a^2)^11. This problem involves key algebraic concepts and the binomial theorem. Step-by-step solutions provided. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation