The largest term in the expansion of `((b)/(2) + (b)/(2))^(100) ` is

Options

A. `b^(100)`
B. `((b)/(2))^(100)`
C. `""^(100)C_(50)((b)/(2))^(100)`
D. `""^(100)C_(50)b^(100)`

Explanation

To find the largest term in the expansion of \(\left(\frac{b}{2} + \frac{b}{2}\right)^{100}\), we can follow these steps:

### Step 1: Identify the expression
The expression can be simplified as:
\[ \left(\frac{b}{2} + \frac{b}{2}\right)^{100} = \left(b\right)^{100} \left(\frac{1}{2} + \frac{1}{2}\right)^{100} = b^{100} \]
However, since we are looking for the largest term in the binomial expansion, we will consider the general form of the binomial theorem.

### Step 2: Apply the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, \(a = \frac{b}{2}\) and \(b = \frac{b}{2}\), and \(n = 100\). Therefore, the expansion will be:
\[ \left(\frac{b}{2} + \frac{b}{2}\right)^{100} = \sum_{k=0}^{100} \binom{100}{k} \left(\frac{b}{2}\right)^{100-k} \left(\frac{b}{2}\right)^{k} \]

### Step 3: Determine the general term
The general term \(T_k\) in the expansion is given by:
\[ T_k = \binom{100}{k} \left(\frac{b}{2}\right)^{100} = \binom{100}{k} \frac{b^{100}}{2^{100}} \]

### Step 4: Find the largest term
To find the largest term, we need to identify the term where \(k\) is approximately equal to \(\frac{n}{2}\). Since \(n = 100\), the middle term occurs at \(k = 50\).

### Step 5: Calculate the middle term
The middle term \(T_{50}\) is:
\[ T_{50} = \binom{100}{50} \left(\frac{b}{2}\right)^{100} = \binom{100}{50} \frac{b^{100}}{2^{100}} \]

### Final Result
Thus, the largest term in the expansion is:
\[ T_{50} = \binom{100}{50} \frac{b^{100}}{2^{100}} \]