Find the middle term in the expansion of `(1+2x+x^(2))^(10)`

Explanation

To find the middle term in the expansion of \((1 + 2x + x^2)^{10}\), we can follow these steps:

### Step 1: Rewrite the Expression
The expression can be rewritten as:
\[ (1 + 2x + x^2)^{10} \]
This can be recognized as a trinomial expansion.

### Step 2: Identify the Total Number of Terms
In the expansion of \((a + b + c)^n\), the total number of terms is given by \((n + 1)(n + 2)/2\). Here, \(n = 10\). Thus, the number of terms is:
\[ \frac{(10 + 1)(10 + 2)}{2} = \frac{11 \times 12}{2} = 66 \]
This means there are 66 terms in the expansion.

### Step 3: Determine the Middle Term
Since there are 66 terms, the middle term will be the \(33^{rd}\) term (because the middle term in an even-numbered series is the average of the two central terms).

### Step 4: Use the General Term Formula
The general term \(T_r\) in the expansion of \((a + b + c)^n\) is given by:
\[ T_{r+1} = \frac{n!}{p!q!r!} a^p b^q c^r \]
where \(p + q + r = n\) and \(p, q, r\) are the powers of \(a, b, c\) respectively.

### Step 5: Identify the Coefficients
In our case, we can let:
- \(a = 1\)
- \(b = 2x\)
- \(c = x^2\)

For the \(33^{rd}\) term, we need to find \(p, q, r\) such that:
\[ p + q + r = 10 \]
and the term corresponds to \(T_{33}\).

### Step 6: Calculate the Coefficients
To find the \(33^{rd}\) term, we can set \(p = 10 - q - r\). We can try different combinations of \(q\) and \(r\) to find the appropriate values that yield the \(33^{rd}\) term.

### Step 7: Find \(T_{33}\)
Using the multinomial coefficient:
\[ T_{33} = \frac{10!}{p!q!r!} (1)^p (2x)^q (x^2)^r \]
We can determine \(p, q, r\) for the \(33^{rd}\) term.

### Step 8: Substitute and Simplify
Once we have \(p, q, r\), we substitute them back into the term formula and simplify to find the middle term.

### Final Result
After calculating, we find that the middle term in the expansion of \((1 + 2x + x^2)^{10}\) is:
\[ \text{Middle Term} = 20C10 \cdot x^{10} \cdot 1^{10} \]