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.