Find the numerically greatest terms in the expansion of ` (3y + 7x)^10` when ` x = 1/3, y = 1/2 `

Answer

To find the numerically greatest term in the expansion of \( (3y + 7x)^{10} \) when \( x = \frac{1}{3} \) and \( y = \frac{1}{2} \), we will follow these steps:

### Step 1: Identify the parameters
The expression we are working with is \( (3y + 7x)^{10} \). Here, \( n = 10 \).

### Step 2: Rewrite the expression
We can rewrite the expression in a form similar to \( (a + b)^n \):
\[ (3y + 7x)^{10} = 3^{10}y^{10} \left(1 + \frac{7x}{3y}\right)^{10} \]

### Step 3: Calculate \( \frac{7x}{3y} \)
Substituting the values of \( x \) and \( y \):
\[ \frac{7x}{3y} = \frac{7 \cdot \frac{1}{3}}{3 \cdot \frac{1}{2}} = \frac{\frac{7}{3}}{\frac{3}{2}} = \frac{7}{3} \cdot \frac{2}{3} = \frac{14}{9} \]

### Step 4: Find \( m \)
To find the term that is numerically greatest, we use the formula:
\[ m = \frac{n + 1}{1 + \frac{7x}{3y}} = \frac{10 + 1}{1 + \frac{14}{9}} = \frac{11}{\frac{23}{9}} = \frac{11 \cdot 9}{23} = \frac{99}{23} \approx 4.304 \]

### Step 5: Determine the integer part of \( m \)
Since \( m \) is not an integer, we take the greatest integer less than \( m \):
\[ \lfloor m \rfloor = 4 \]

### Step 6: Identify the term
The numerically greatest term corresponds to \( T_{m + 1} \), which is \( T_5 \).

### Step 7: Calculate \( T_5 \)
Using the general term formula for binomial expansion:
\[ T_r = \binom{n}{r} (3y)^{n-r} (7x)^r \]
For \( T_5 \) (where \( r = 5 \)):
\[ T_5 = \binom{10}{5} (3y)^{10-5} (7x)^5 = \binom{10}{5} (3y)^5 (7x)^5 \]

### Step 8: Substitute values of \( x \) and \( y \)
Now substituting \( y = \frac{1}{2} \) and \( x = \frac{1}{3} \):
\[ T_5 = \binom{10}{5} (3 \cdot \frac{1}{2})^5 (7 \cdot \frac{1}{3})^5 = \binom{10}{5} \left(\frac{3}{2}\right)^5 \left(\frac{7}{3}\right)^5 \]

### Step 9: Simplify
Calculating \( T_5 \):
\[ T_5 = \binom{10}{5} \left(\frac{3}{2} \cdot \frac{7}{3}\right)^5 = \binom{10}{5} \left(\frac{7}{2}\right)^5 \]

### Step 10: Final Calculation
Calculating \( \binom{10}{5} = 252 \):
\[ T_5 = 252 \cdot \left(\frac{7}{2}\right)^5 = 252 \cdot \frac{16807}{32} \]

Thus, the numerically greatest term in the expansion of \( (3y + 7x)^{10} \) when \( x = \frac{1}{3} \) and \( y = \frac{1}{2} \) is:
\[ T_5 = \frac{252 \cdot 16807}{32} \]