Find the numerically greatest terms in the expansion of `(3 + (2x)/(5))^12` when x = 3/4`

Answer

To find the numerically greatest term in the expansion of \((3 + \frac{2x}{5})^{12}\) when \(x = \frac{3}{4}\), we will follow these steps:

### Step 1: Identify the values of \(n\) and \(x\)
The expression is \((3 + \frac{2x}{5})^{12}\). Here, \(n = 12\) and we will substitute \(x = \frac{3}{4}\) later.

### Step 2: Rewrite the expression
We can factor out \(3^{12}\) from the expression:
\[ (3 + \frac{2x}{5})^{12} = 3^{12} \left(1 + \frac{2x}{15}\right)^{12} \]

### Step 3: Calculate \(\frac{2x}{15}\)
Substituting \(x = \frac{3}{4}\):
\[ \frac{2x}{15} = \frac{2 \cdot \frac{3}{4}}{15} = \frac{3}{30} = \frac{1}{10} \]

### Step 4: Find \(m\)
Using the formula \(m = \frac{n + 1 \cdot |x|}{|x| + 1}\):
- Here, \(|x| = \frac{1}{10}\).
- So, \(m = \frac{12 + 1 \cdot \frac{1}{10}}{\frac{1}{10} + 1} = \frac{12 + \frac{1}{10}}{\frac{1}{10} + 1} = \frac{12.1}{1.1}\).

### Step 5: Simplify \(m\)
Calculating \(m\):
\[ m = \frac{121}{11} = 11 \]

### Step 6: Determine the greatest term
Since \(m\) is an integer, the numerically greatest terms are \(T_m\) and \(T_{m+1}\), which correspond to \(T_{11}\) and \(T_{12}\).

### Step 7: Find \(T_{11}\) and \(T_{12}\)
The general term \(T_r\) in the binomial expansion is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case:
- \(a = 3\)
- \(b = \frac{2x}{5} = \frac{2 \cdot \frac{3}{4}}{5} = \frac{3}{10}\)

#### For \(T_{11}\):
\[ T_{11} = \binom{12}{10} \cdot 3^{12-10} \cdot \left(\frac{3}{10}\right)^{10} = \binom{12}{2} \cdot 3^2 \cdot \left(\frac{3}{10}\right)^{10} \]

Calculating:
\[ \binom{12}{2} = \frac{12 \cdot 11}{2} = 66 \]
\[ T_{11} = 66 \cdot 9 \cdot \left(\frac{3^{10}}{10^{10}}\right) = 594 \cdot \frac{59049}{10000000000} \]

#### For \(T_{12}\):
\[ T_{12} = \binom{12}{11} \cdot 3^{12-11} \cdot \left(\frac{3}{10}\right)^{11} = 12 \cdot 3^1 \cdot \left(\frac{3}{10}\right)^{11} \]
Calculating:
\[ T_{12} = 12 \cdot 3 \cdot \frac{3^{11}}{10^{11}} = 36 \cdot \frac{177147}{100000000000} \]

### Conclusion
The numerically greatest terms in the expansion of \((3 + \frac{2x}{5})^{12}\) when \(x = \frac{3}{4}\) are \(T_{11}\) and \(T_{12}\).