Find the general term in the expansion of `(1- (5x)/(3))^(-3)`

Answer

To find the general term in the expansion of \( \left(1 - \frac{5x}{3}\right)^{-3} \), we will use the Binomial Theorem. Let's go through the steps systematically.

### Step 1: Identify the parameters
We start with the expression:
\[ \left(1 - \frac{5x}{3}\right)^{-3} \]
In the standard form of the binomial expansion \( (1 - x)^{-n} \), we can identify:
- \( x = \frac{5x}{3} \)
- \( n = 3 \)

### Step 2: Write the formula for the general term
The general term \( T_{r+1} \) in the expansion of \( (1 - x)^{-n} \) is given by:
\[ T_{r+1} = \binom{n+r-1}{r} (-x)^r \]
For our case, since \( n = 3 \), the formula becomes:
\[ T_{r+1} = \binom{3+r-1}{r} \left(-\frac{5x}{3}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{r+2}{r} \left(-\frac{5x}{3}\right)^r \]

### Step 3: Simplify the binomial coefficient
The binomial coefficient \( \binom{r+2}{r} \) can also be expressed as:
\[ \binom{r+2}{r} = \frac{(r+2)!}{r! \cdot 2!} = \frac{(r+2)(r+1)}{2} \]

### Step 4: Substitute back into the general term
Substituting the binomial coefficient back into the general term, we have:
\[ T_{r+1} = \frac{(r+2)(r+1)}{2} \left(-\frac{5x}{3}\right)^r \]

### Step 5: Final expression for the general term
Thus, the general term in the expansion of \( \left(1 - \frac{5x}{3}\right)^{-3} \) is:
\[ T_{r+1} = \frac{(r+2)(r+1)}{2} \left(-\frac{5x}{3}\right)^r \]

### Summary
The general term \( T_{r+1} \) in the expansion of \( \left(1 - \frac{5x}{3}\right)^{-3} \) is:
\[ T_{r+1} = \frac{(r+2)(r+1)}{2} \left(-\frac{5x}{3}\right)^r \]