If the pth, qth and rt terms of an A.P. be x,y,z respectively show that:
`x(q-r)+y(r-p)+z(p-q)=0`

Answer

To prove that \( x(q - r) + y(r - p) + z(p - q) = 0 \) given that \( x, y, z \) are the \( p \)-th, \( q \)-th, and \( r \)-th terms of an arithmetic progression (A.P.), we can follow these steps:

### Step 1: Define the terms of the A.P.
Let the first term of the A.P. be \( a \) and the common difference be \( d \). Then, the terms can be expressed as:
- The \( p \)-th term \( x = a + (p - 1)d \)
- The \( q \)-th term \( y = a + (q - 1)d \)
- The \( r \)-th term \( z = a + (r - 1)d \)

### Step 2: Substitute the terms into the equation
We need to substitute \( x, y, z \) into the equation \( x(q - r) + y(r - p) + z(p - q) \):
\[ x(q - r) + y(r - p) + z(p - q) = (a + (p - 1)d)(q - r) + (a + (q - 1)d)(r - p) + (a + (r - 1)d)(p - q) \]

### Step 3: Expand the equation
Now we expand each term:
1. For \( x(q - r) \):
\[ x(q - r) = (a + (p - 1)d)(q - r) = a(q - r) + (p - 1)d(q - r) \]
2. For \( y(r - p) \):
\[ y(r - p) = (a + (q - 1)d)(r - p) = a(r - p) + (q - 1)d(r - p) \]
3. For \( z(p - q) \):
\[ z(p - q) = (a + (r - 1)d)(p - q) = a(p - q) + (r - 1)d(p - q) \]

### Step 4: Combine all the terms
Combining all the expanded terms gives:
\[ a(q - r) + a(r - p) + a(p - q) + (p - 1)d(q - r) + (q - 1)d(r - p) + (r - 1)d(p - q) \]

### Step 5: Simplify the equation
Now, we can group the \( a \) terms:
\[ a[(q - r) + (r - p) + (p - q)] + [(p - 1)d(q - r) + (q - 1)d(r - p) + (r - 1)d(p - q)] \]
The first part simplifies to:
\[ a[0] = 0 \]
Now, we need to simplify the second part:
\[ (p - 1)d(q - r) + (q - 1)d(r - p) + (r - 1)d(p - q) \]

### Step 6: Factor out \( d \)
Factoring out \( d \):
\[ d[(p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q)] \]

### Step 7: Show that the expression equals zero
The expression inside the brackets can be shown to equal zero:
1. Expanding it gives:
\[ (pq - pr - q + r) + (qr - qp - r + p) + (rp - rq - p + q) \]
2. Collecting like terms leads to:
\[ pq - pr + qr - qp + rp - rq - q + r - r + p - p + q = 0 \]

Thus, we conclude that:
\[ x(q - r) + y(r - p) + z(p - q) = 0 \]

### Final Result
We have shown that:
\[ x(q - r) + y(r - p) + z(p - q) = 0 \]