If the coefficent of `(r-5)th and (2r-1)th` terms in the expansion of `(1+x)^(34)` are equal , find the value of r.

Explanation

To solve the problem, we need to find the value of \( r \) such that the coefficients of the \( (r-5) \)th term and the \( (2r-1) \)th term in the expansion of \( (1+x)^{34} \) are equal.

### Step 1: Identify the General Term
The general term \( T_{r+1} \) in the expansion of \( (1+x)^{n} \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For our case, \( n = 34 \), \( a = 1 \), and \( b = x \). Thus, the general term becomes:
\[ T_{r+1} = \binom{34}{r} (1)^{34-r} x^r = \binom{34}{r} x^r \]

### Step 2: Coefficient of the \( (r-5) \)th Term
The \( (r-5) \)th term corresponds to \( r-5 \), so we need to find:
\[ \text{Coefficient of } T_{r-5} = \binom{34}{r-5} \]

### Step 3: Coefficient of the \( (2r-1) \)th Term
The \( (2r-1) \)th term corresponds to \( 2r-1 \), so we need to find:
\[ \text{Coefficient of } T_{2r-1} = \binom{34}{2r-1} \]

### Step 4: Set the Coefficients Equal
According to the problem, these coefficients are equal:
\[ \binom{34}{r-5} = \binom{34}{2r-1} \]

### Step 5: Use the Property of Binomial Coefficients
The property of binomial coefficients states that:
\[ \binom{n}{k} = \binom{n}{n-k} \]
This gives us two equations to solve:
1. \( r - 5 = 2r - 1 \)
2. \( r - 5 = 34 - (2r - 1) \)

### Step 6: Solve the First Equation
From \( r - 5 = 2r - 1 \):
\[ r - 5 = 2r - 1 \implies -5 + 1 = 2r - r \implies -4 = r \implies r = -4 \quad \text{(not valid, since } r \text{ must be a natural number)} \]

### Step 7: Solve the Second Equation
From \( r - 5 = 34 - (2r - 1) \):
\[ r - 5 = 34 - 2r + 1 \implies r - 5 = 35 - 2r \implies r + 2r = 35 + 5 \implies 3r = 40 \implies r = \frac{40}{3} \quad \text{(not valid)} \]

### Step 8: Solve the Valid Equation
Now, we can also consider:
\[ 2r - 1 = 34 - (r - 5) \implies 2r - 1 = 34 - r + 5 \implies 2r + r = 39 \implies 3r = 39 \implies r = 13 \]

### Conclusion
The value of \( r \) that satisfies the condition is:
\[ \boxed{13} \]