The given image refers to finding the middle term in the expansion of $(1+x)^{2n}$ using the binomial theorem. Here is the explanation:

Key Details:

1. The expansion of $(1 + x)^{2n}$ contains $2n + 1$ terms.
2. When $2n$ is even, the middle term is the $(n+1)$-th term.
3. The $(n+1)$-th term in the binomial expansion is generally given as: $T_{n+1} = \binom{2n}{n} x^n$ where $\binom{2n}{n}$ is the binomial coefficient.

This approach ensures the middle term is correctly identified in the expansion.

Would you like additional explanations or an example worked out for this?

Questions to Explore:

1. How is the binomial coefficient $\binom{2n}{n}$ calculated?
2. Why does the number of terms in the expansion of $(1+x)^{2n}$ depend on $2n$?
3. What happens when $2n$ is odd? Is the middle term defined differently?
4. Can the same logic apply to expansions of $(1-x)^{2n}$?
5. How is the result used in approximations or numerical problems?

Tip:

Always confirm whether the power of the binomial is even or odd before determining the middle term.