The greatest term in the expansion of `(3+2x)^(51)`, where `x=(1)/(5)`, is

Options

A. `5^(th)` term
B. `6^(th)` term
C. `7^(th)` term
D. `9^(th)` term

Answer

To find the greatest term in the expansion of \((3 + 2x)^{51}\) where \(x = \frac{1}{5}\), we can follow these steps:

### Step 1: Identify the values of \(x\) and \(y\)
In the expression \((3 + 2x)^{51}\), we can identify:
- \(x = 3\)
- \(y = 2x = 2 \cdot \frac{1}{5} = \frac{2}{5}\)

### Step 2: Use the formula for the greatest term
The greatest term in the binomial expansion of \((x + y)^n\) occurs at the term \(T_r\), where:
\[ r = \left\lfloor \frac{n + 1}{2} \cdot \frac{y}{x + y} \right\rfloor \]
In our case, \(n = 51\), \(x = 3\), and \(y = \frac{2}{5}\).

### Step 3: Calculate \(r\)
First, we need to calculate:
\[ \frac{y}{x + y} = \frac{\frac{2}{5}}{3 + \frac{2}{5}} = \frac{\frac{2}{5}}{\frac{15}{5} + \frac{2}{5}} = \frac{\frac{2}{5}}{\frac{17}{5}} = \frac{2}{17} \]
Now substituting this back into the formula for \(r\):
\[ r = \left\lfloor \frac{51 + 1}{2} \cdot \frac{2}{17} \right\rfloor = \left\lfloor \frac{52}{2} \cdot \frac{2}{17} \right\rfloor = \left\lfloor 26 \cdot \frac{2}{17} \right\rfloor = \left\lfloor \frac{52}{17} \right\rfloor \]
Calculating \(\frac{52}{17}\):
\[ \frac{52}{17} \approx 3.0588 \]
Thus, \(r = 3\).

### Step 4: Identify the term \(T_r\)
The \(r\)-th term in the expansion is given by:
\[ T_r = \binom{n}{r} x^{n-r} y^r \]
Substituting \(n = 51\), \(r = 3\), \(x = 3\), and \(y = \frac{2}{5}\):
\[ T_3 = \binom{51}{3} (3)^{51-3} \left(\frac{2}{5}\right)^3 \]

### Step 5: Calculate the binomial coefficient and powers
Calculating \(\binom{51}{3}\):
\[ \binom{51}{3} = \frac{51 \cdot 50 \cdot 49}{3 \cdot 2 \cdot 1} = 23425 \]
Calculating \( (3)^{48} \) and \( \left(\frac{2}{5}\right)^3 \):
\[ \left(\frac{2}{5}\right)^3 = \frac{8}{125} \]
Thus,
\[ T_3 = 23425 \cdot (3^{48}) \cdot \frac{8}{125} \]

### Step 6: Final calculation
To find the greatest term, we can compute:
\[ T_3 = \frac{23425 \cdot 8 \cdot 3^{48}}{125} \]

### Conclusion
The greatest term in the expansion of \((3 + 2x)^{51}\) where \(x = \frac{1}{5}\) is:
\[ T_3 = \frac{23425 \cdot 8 \cdot 3^{48}}{125} \]