The sum of three numbers in AP is 15 and sum of their squares is 93 find the greatest number

Options

A. 6
B. 10
C. 8
D. 7

Answer

To solve the problem, we need to find three numbers in arithmetic progression (AP) whose sum is 15 and the sum of their squares is 93. Let's denote the three numbers as \( a - d \), \( a \), and \( a + d \).

### Step 1: Set up the equations
1. The sum of the three numbers:
\[ (a - d) + a + (a + d) = 15 \]
Simplifying this gives:
\[ 3a = 15 \]

### Step 2: Solve for \( a \)
2. Dividing both sides by 3:
\[ a = \frac{15}{3} = 5 \]

### Step 3: Set up the second equation for the sum of squares
3. The sum of the squares of the three numbers:
\[ (a - d)^2 + a^2 + (a + d)^2 = 93 \]
Expanding this:
\[ (a^2 - 2ad + d^2) + a^2 + (a^2 + 2ad + d^2) = 93 \]
Combining like terms:
\[ 3a^2 + 2d^2 = 93 \]

### Step 4: Substitute \( a \) into the equation
4. Substitute \( a = 5 \):
\[ 3(5^2) + 2d^2 = 93 \]
This simplifies to:
\[ 3(25) + 2d^2 = 93 \]
\[ 75 + 2d^2 = 93 \]

### Step 5: Solve for \( d^2 \)
5. Rearranging gives:
\[ 2d^2 = 93 - 75 \]
\[ 2d^2 = 18 \]
Dividing both sides by 2:
\[ d^2 = 9 \]

### Step 6: Find \( d \)
6. Taking the square root:
\[ d = \pm 3 \]

### Step 7: Find the three numbers
7. Now we can find the three numbers:
- If \( d = 3 \):
- The numbers are \( 5 - 3 = 2 \), \( 5 \), and \( 5 + 3 = 8 \).
- If \( d = -3 \):
- The numbers are \( 5 - (-3) = 8 \), \( 5 \), and \( 5 + (-3) = 2 \).

### Step 8: Identify the greatest number
8. In both cases, the greatest number is \( 8 \).

### Final Answer
Thus, the greatest number is \( \boxed{8} \).

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