Find the numerically greatest term (s) in the expansion of
`(3+7x)^(n)" when "x=(4)/(5),n=15`

Answer

To find the numerically greatest term in the expansion of \((3 + 7x)^{15}\) when \(x = \frac{4}{5}\), we can follow these steps:

### Step 1: Identify the Binomial Expansion
The binomial expansion of \((a + b)^n\) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 3\), \(b = 7x\), and \(n = 15\).

### Step 2: Substitute the Value of \(x\)
We substitute \(x = \frac{4}{5}\) into the expression:
\[ b = 7x = 7 \cdot \frac{4}{5} = \frac{28}{5} \]

### Step 3: Find the Ratio for m
To find the term that is numerically greatest, we need to calculate:
\[ m = \frac{(n + 1) \cdot |b/a|}{|b/a| + 1} \]
Here, \(n = 15\), \(a = 3\), and \(b = \frac{28}{5}\):
\[ \frac{b}{a} = \frac{\frac{28}{5}}{3} = \frac{28}{15} \]

### Step 4: Calculate m
Now we can substitute into the formula:
\[ m = \frac{(15 + 1) \cdot \frac{28}{15}}{\frac{28}{15} + 1} = \frac{16 \cdot \frac{28}{15}}{\frac{28}{15} + \frac{15}{15}} = \frac{16 \cdot \frac{28}{15}}{\frac{43}{15}} = \frac{16 \cdot 28}{43} \]
Calculating this gives:
\[ m = \frac{448}{43} \approx 10.41 \]

### Step 5: Determine the Greatest Integer
Since \(m\) is not an integer, we take the greatest integer less than \(m\) and add 1:
\[ \lfloor m \rfloor + 1 = 10 + 1 = 11 \]
Thus, the numerically greatest term corresponds to \(T_{11}\).

### Step 6: Find the 11th Term
The 11th term in the expansion is given by:
\[ T_{11} = \binom{15}{10} \cdot 3^{15-10} \cdot (7x)^{10} \]
Calculating this:
\[ T_{11} = \binom{15}{10} \cdot 3^5 \cdot \left(7 \cdot \frac{4}{5}\right)^{10} \]
Calculating \(\binom{15}{10}\):
\[ \binom{15}{10} = \binom{15}{5} = \frac{15 \cdot 14 \cdot 13 \cdot 12 \cdot 11}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 3003 \]
Calculating \(3^5\):
\[ 3^5 = 243 \]
Calculating \((7 \cdot \frac{4}{5})^{10}\):
\[ (7 \cdot \frac{4}{5})^{10} = \left(\frac{28}{5}\right)^{10} = \frac{28^{10}}{5^{10}} \]
Putting it all together:
\[ T_{11} = 3003 \cdot 243 \cdot \frac{28^{10}}{5^{10}} \]

### Final Result
The numerically greatest term in the expansion of \((3 + 7x)^{15}\) when \(x = \frac{4}{5}\) is:
\[ T_{11} = 3003 \cdot 243 \cdot \frac{28^{10}}{5^{10}} \]