If the coefficients of rth and `(r+1)t h` terms in the expansion of `(3+7x)^(29)` are equal, then `r` equals a. 15 b. 21 c. 14 d. none of these

Options

A. 15
B. 21
C. 14
D. none of these

Explanation

To solve the problem, we need to find the value of \( r \) such that the coefficients of the \( r \)-th and \( (r+1) \)-th terms in the expansion of \( (3 + 7x)^{29} \) are equal.

### Step-by-step Solution:

1. **Understanding the Binomial Expansion**:
The binomial expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1} \]
where \( T_k \) is the \( k \)-th term, \( n \) is the power, \( a \) and \( b \) are the terms in the binomial.

2. **Identifying the Terms**:
For our case, \( a = 3 \), \( b = 7x \), and \( n = 29 \).
- The \( r \)-th term \( T_{r} \) is given by:
\[ T_r = \binom{29}{r-1} \cdot 3^{29-(r-1)} \cdot (7x)^{r-1} \]
- The \( (r+1) \)-th term \( T_{r+1} \) is given by:
\[ T_{r+1} = \binom{29}{r} \cdot 3^{29-r} \cdot (7x)^{r} \]

3. **Extracting the Coefficients**:
- Coefficient of \( T_r \):
\[ C_r = \binom{29}{r-1} \cdot 3^{30-r} \cdot 7^{r-1} \]
- Coefficient of \( T_{r+1} \):
\[ C_{r+1} = \binom{29}{r} \cdot 3^{29-r} \cdot 7^{r} \]

4. **Setting the Coefficients Equal**:
Since the coefficients are equal:
\[ \binom{29}{r-1} \cdot 3^{30-r} \cdot 7^{r-1} = \binom{29}{r} \cdot 3^{29-r} \cdot 7^{r} \]

5. **Simplifying the Equation**:
Dividing both sides by \( 3^{29-r} \) and \( 7^{r-1} \):
\[ \binom{29}{r-1} \cdot 3 = \binom{29}{r} \cdot 7 \]

6. **Using the Property of Binomial Coefficients**:
We know that:
\[ \binom{29}{r} = \frac{29!}{r!(29-r)!} \quad \text{and} \quad \binom{29}{r-1} = \frac{29!}{(r-1)!(30-r)!} \]
Thus, substituting these into the equation gives:
\[ \frac{29!}{(r-1)!(30-r)!} \cdot 3 = \frac{29!}{r!(29-r)!} \cdot 7 \]

7. **Cancelling Factorials**:
Cancelling \( 29! \) from both sides:
\[ \frac{3}{(30-r)} = \frac{7}{r} \]

8. **Cross Multiplying**:
Cross multiplying gives:
\[ 3r = 7(30 - r) \]
Expanding this:
\[ 3r = 210 - 7r \]

9. **Solving for \( r \)**:
Combining like terms:
\[ 3r + 7r = 210 \implies 10r = 210 \implies r = \frac{210}{10} = 21 \]

### Conclusion:
Thus, the value of \( r \) is \( 21 \).