Find `a` and `b` such that the numbers `a, 9, b, 25` form an AP.

Options

A. 17 and 1
B. 1 and 17
C. 2 and 15
D. none of these

Explanation

To find the values of `a` and `b` such that the numbers `a, 9, b, 25` form an Arithmetic Progression (AP), we can follow these steps:

### Step 1: Understand the property of AP
In an AP, the difference between consecutive terms is constant. This means that the difference between the second term and the first term should be equal to the difference between the third term and the second term, and also equal to the difference between the fourth term and the third term.

### Step 2: Set up the equations
Let the common difference be `d`. We can express the terms in terms of `d`:
- The second term: \( 9 = a + d \)
- The third term: \( b = 9 + d \)
- The fourth term: \( 25 = b + d \)

### Step 3: Express `b` in terms of `d`
From the equation for the third term, we can write:
\[ b = 9 + d \]

### Step 4: Substitute `b` in the fourth term equation
Now substitute `b` into the fourth term equation:
\[ 25 = (9 + d) + d \]
This simplifies to:
\[ 25 = 9 + 2d \]

### Step 5: Solve for `d`
Rearranging gives:
\[ 2d = 25 - 9 \]
\[ 2d = 16 \]
\[ d = 8 \]

### Step 6: Find `b`
Now that we have `d`, we can find `b`:
\[ b = 9 + d = 9 + 8 = 17 \]

### Step 7: Find `a`
Next, we can find `a` using the first equation:
\[ 9 = a + d \]
Substituting the value of `d`:
\[ 9 = a + 8 \]
Rearranging gives:
\[ a = 9 - 8 = 1 \]

### Final Answer
Thus, the values of `a` and `b` are:
- \( a = 1 \)
- \( b = 17 \)