The coefficients of `(2r +1)`th and `(r+2)`th terms in the expansions of `(1 +x)^(43)` are equal. Find the value of `r`.

Answer

To solve the problem, we need to find the value of \( r \) such that the coefficients of the \( (2r + 1) \)th term and the \( (r + 2) \)th term in the expansion of \( (1 + x)^{43} \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term in the Binomial Expansion**:
The general term \( T_k \) in the expansion of \( (1 + x)^n \) is given by:
\[ T_k = \binom{n}{k-1} x^{k-1} \]
where \( n = 43 \) in this case.

2. **Find the Coefficient of the \( (2r + 1) \)th Term**:
The \( (2r + 1) \)th term corresponds to \( k = 2r + 1 \). Therefore, the coefficient is:
\[ \text{Coefficient of } T_{2r + 1} = \binom{43}{2r} \]

3. **Find the Coefficient of the \( (r + 2) \)th Term**:
The \( (r + 2) \)th term corresponds to \( k = r + 2 \). Therefore, the coefficient is:
\[ \text{Coefficient of } T_{r + 2} = \binom{43}{r + 1} \]

4. **Set the Coefficients Equal**:
According to the problem, these coefficients are equal:
\[ \binom{43}{2r} = \binom{43}{r + 1} \]

5. **Apply the Property of Binomial Coefficients**:
The equality \( \binom{n}{x} = \binom{n}{y} \) holds if either:
- \( x + y = n \)
- \( x = y \)

We will apply both conditions.

6. **Condition 1: \( 2r + (r + 1) = 43 \)**:
\[ 3r + 1 = 43 \]
\[ 3r = 42 \quad \Rightarrow \quad r = 14 \]

7. **Condition 2: \( 2r = r + 1 \)**:
\[ 2r - r = 1 \quad \Rightarrow \quad r = 1 \]

8. **Check Validity of Solutions**:
Since \( r \) must be a non-negative integer, we have two potential solutions: \( r = 14 \) and \( r = 1 \). However, we need to check if both are valid in the context of the problem.

- For \( r = 14 \):
- \( 2r + 1 = 29 \)
- \( r + 2 = 16 \)
- Coefficients: \( \binom{43}{28} = \binom{43}{16} \) (valid)

- For \( r = 1 \):
- \( 2r + 1 = 3 \)
- \( r + 2 = 3 \)
- Coefficients: \( \binom{43}{2} = \binom{43}{2} \) (valid)

9. **Final Answer**:
The valid values of \( r \) are \( 14 \) and \( 1 \). However, since the problem asks for a specific value, we can conclude:
\[ \text{The value of } r = 14. \]