Let `A_1, A_2`,.......`A_n` be the n points on the straight-line y = px + q. The coordinates of `A_k` is `(x_k, y_k)`, where k = 1,2,.... n such that `x_1, x_2`, ......, `x_n` are in arithmetic progression. The coordinates of` A_2` is (2, –2) and` A_24` is (68, 31).
The y-ordinates of `A_8` is

Options

A. 13
B. 10
C. 7
D. 5.5

Answer

To find the y-coordinate of point \( A_8 \) on the line defined by the equation \( y = px + q \), we start by using the information given in the problem.

### Step 1: Identify the coordinates of \( A_2 \) and \( A_{24} \)
We know:
- \( A_2 = (2, -2) \)
- \( A_{24} = (68, 31) \)

### Step 2: Express the y-coordinates in terms of an arithmetic progression
Let the y-coordinates of the points \( A_k \) be represented as:
- \( y_k = a + (k - 1)d \)
where \( a \) is the first term (y-coordinate of \( A_1 \)) and \( d \) is the common difference.

### Step 3: Write equations for \( A_2 \) and \( A_{24} \)
From the coordinates of \( A_2 \):
\[ y_2 = a + (2 - 1)d = a + d = -2 \quad \text{(1)} \]

From the coordinates of \( A_{24} \):
\[ y_{24} = a + (24 - 1)d = a + 23d = 31 \quad \text{(2)} \]

### Step 4: Solve the system of equations
We have two equations:
1. \( a + d = -2 \)
2. \( a + 23d = 31 \)

Subtract equation (1) from equation (2):
\[ (a + 23d) - (a + d) = 31 - (-2) \]
This simplifies to:
\[ 22d = 33 \implies d = \frac{33}{22} = \frac{3}{2} \]

### Step 5: Substitute \( d \) back to find \( a \)
Substituting \( d \) back into equation (1):
\[ a + \frac{3}{2} = -2 \]
\[ a = -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2} \]

### Step 6: Find the y-coordinate of \( A_8 \)
Now we can find the y-coordinate of \( A_8 \):
\[ y_8 = a + (8 - 1)d = a + 7d \]
Substituting the values of \( a \) and \( d \):
\[ y_8 = -\frac{7}{2} + 7 \cdot \frac{3}{2} \]
Calculating \( 7 \cdot \frac{3}{2} \):
\[ 7 \cdot \frac{3}{2} = \frac{21}{2} \]
Now substituting:
\[ y_8 = -\frac{7}{2} + \frac{21}{2} = \frac{21 - 7}{2} = \frac{14}{2} = 7 \]

### Final Answer
The y-coordinate of \( A_8 \) is \( 7 \).

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