The greatest term in the expansion of
`(1 + 3x)^(54)` when ` x = (1)/(3)`,is

Options

A. `28^(th)`
B. `25^(th)`
C. `26^(th)`
D. `24^(th)`

Explanation

To find the greatest term in the expansion of \((1 + 3x)^{54}\) when \(x = \frac{1}{3}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \(T_r\) in the expansion of \((1 + 3x)^{54}\) is given by:
\[ T_r = \binom{54}{r} (3x)^r \]
where \(r\) is the term index starting from 0.

2. **Substitute \(x\)**:
We substitute \(x = \frac{1}{3}\) into the general term:
\[ T_r = \binom{54}{r} \left(3 \cdot \frac{1}{3}\right)^r = \binom{54}{r} (1)^r = \binom{54}{r} \]

3. **Find the Ratio of Consecutive Terms**:
To find the greatest term, we consider the ratio of consecutive terms \(T_{r+1}\) and \(T_r\):
\[ \frac{T_{r+1}}{T_r} = \frac{\binom{54}{r+1}}{\binom{54}{r}} = \frac{54 - r}{r + 1} \]

4. **Set Up the Inequality**:
For \(T_{r+1}\) to be greater than \(T_r\), we need:
\[ \frac{T_{r+1}}{T_r} > 1 \implies \frac{54 - r}{r + 1} > 1 \]

5. **Solve the Inequality**:
Solving the inequality:
\[ 54 - r > r + 1 \implies 54 - 1 > 2r \implies 53 > 2r \implies r < \frac{53}{2} = 26.5 \]
Thus, the largest integer \(r\) can take is \(26\).

6. **Identify the Greatest Term**:
The greatest term corresponds to \(T_{r+1}\), which is \(T_{27}\) (since \(r\) starts from 0).

7. **Final Result**:
Therefore, the greatest term in the expansion of \((1 + 3x)^{54}\) when \(x = \frac{1}{3}\) is the **27th term**.