Find numerically greatest term in the expansion of `(5-3x)^(7)` when x=2/3

Answer

To find the numerically greatest term in the expansion of \((5 - 3x)^7\) when \(x = \frac{2}{3}\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \(T_{r+1}\) in the binomial expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
In our case, \(a = 5\), \(b = -3x\), and \(n = 7\).

2. **Determine the Value of \(r\)**:
To find the term that is numerically greatest, we use the formula for \(r\):
\[ r = \left\lfloor \frac{n + 1}{2} \cdot \frac{|a|}{|b| + |a|} \right\rfloor \]
where \(|a| = 5\) and \(|b| = 3x\). First, we need to calculate \(|b|\) when \(x = \frac{2}{3}\):
\[ |b| = 3 \cdot \frac{2}{3} = 2 \]

3. **Substituting Values**:
Now substituting into the formula:
\[ r = \left\lfloor \frac{7 + 1}{2} \cdot \frac{5}{2 + 5} \right\rfloor = \left\lfloor \frac{8}{2} \cdot \frac{5}{7} \right\rfloor = \left\lfloor 4 \cdot \frac{5}{7} \right\rfloor = \left\lfloor \frac{20}{7} \right\rfloor = \left\lfloor 2.857 \right\rfloor = 2 \]

4. **Finding the Greatest Term**:
The term corresponding to \(r = 2\) is \(T_{3}\):
\[ T_{3} = \binom{7}{2} (5)^{7-2} (-3x)^2 \]

5. **Calculating the Coefficient**:
Calculate \(\binom{7}{2}\):
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

6. **Substituting Values**:
Now substitute \(x = \frac{2}{3}\):
\[ T_{3} = 21 \cdot (5)^5 \cdot (-3 \cdot \frac{2}{3})^2 \]
Simplifying \((-3 \cdot \frac{2}{3})^2\):
\[ (-2)^2 = 4 \]

7. **Final Calculation**:
Now we need to compute \(T_{3}\):
\[ T_{3} = 21 \cdot (5^5) \cdot 4 \]
Calculate \(5^5\):
\[ 5^5 = 3125 \]
Thus,
\[ T_{3} = 21 \cdot 3125 \cdot 4 = 21 \cdot 12500 = 26250 \]

### Conclusion:
The numerically greatest term in the expansion of \((5 - 3x)^7\) when \(x = \frac{2}{3}\) is **26250**.