`n` arlithmetic means are inserted between `xa n d2y` and then between `2xa n dydot` If the rth means in each case be equal, then find the ratio `x//ydot`

Explanation

To solve the problem, we need to find the ratio \( \frac{x}{y} \) given that the \( r \)-th arithmetic means inserted between two pairs of numbers are equal. Let's break down the solution step by step.

### Step 1: Define the two series

We have two series:

1. The first series is formed by inserting \( n \) arithmetic means between \( x \) and \( 2y \):
\[ x, a_1, a_2, \ldots, a_n, 2y \]

2. The second series is formed by inserting \( n \) arithmetic means between \( 2x \) and \( y \):
\[ 2x, b_1, b_2, \ldots, b_n, y \]

### Step 2: Find the \( r \)-th arithmetic mean in the first series

The \( r \)-th arithmetic mean in the first series can be calculated using the formula for the \( r \)-th term of an arithmetic progression (AP):
\[ A_r = \frac{x + 2y}{n + 1} \cdot (r) \]
This can be rewritten as:
\[ A_r = x + \frac{r}{n + 1}(2y - x) \]

### Step 3: Find the \( r \)-th arithmetic mean in the second series

Similarly, for the second series, the \( r \)-th arithmetic mean is:
\[ B_r = \frac{2x + y}{n + 1} \cdot (r) \]
This can be rewritten as:
\[ B_r = 2x + \frac{r}{n + 1}(y - 2x) \]

### Step 4: Set the two means equal

According to the problem, the \( r \)-th means from both series are equal:
\[ A_r = B_r \]
Substituting the expressions we derived:
\[ x + \frac{r}{n + 1}(2y - x) = 2x + \frac{r}{n + 1}(y - 2x) \]

### Step 5: Simplify the equation

Rearranging the equation gives:
\[ x + \frac{r(2y - x)}{n + 1} = 2x + \frac{r(y - 2x)}{n + 1} \]
Now, multiply through by \( n + 1 \) to eliminate the denominator:
\[ (n + 1)x + r(2y - x) = (n + 1)2x + r(y - 2x) \]
Expanding both sides:
\[ (n + 1)x + 2ry - rx = 2(n + 1)x + ry - 2rx \]

### Step 6: Collect like terms

Rearranging gives:
\[ (n + 1)x + 2ry - rx - 2(n + 1)x - ry + 2rx = 0 \]
Combining like terms:
\[ (-n - 1 + r)x + (2r - r)y = 0 \]

### Step 7: Solve for the ratio \( \frac{x}{y} \)

Setting the coefficients equal gives:
\[ (-n - 1 + r)x = (r - 2r)y \]
Thus, we can express the ratio:
\[ \frac{x}{y} = \frac{r}{n - r + 1} \]

### Final Answer

The ratio \( \frac{x}{y} \) is:
\[ \frac{x}{y} = \frac{r}{n - r + 1} \]