The sum of three numbers in A.P. 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 step by step, we will use the properties of arithmetic progression (A.P.) and the given conditions.

### Step 1: Define the numbers in A.P.
Let the three numbers in A.P. be:
- First number: \( a - d \)
- Second number: \( a \)
- Third number: \( a + d \)

### Step 2: Use the sum of the numbers
According to the problem, the sum of these three numbers is 15:
\[ (a - d) + a + (a + d) = 15 \]
Simplifying this, we have:
\[ 3a = 15 \]
Thus, we can find \( a \):
\[ a = \frac{15}{3} = 5 \]

### Step 3: Use the sum of the squares
Next, we know that the sum of their squares is 93:
\[ (a - d)^2 + a^2 + (a + d)^2 = 93 \]
Expanding this, we get:
\[ (a^2 - 2ad + d^2) + a^2 + (a^2 + 2ad + d^2) = 93 \]
Combining like terms:
\[ 3a^2 + 2d^2 = 93 \]

### Step 4: Substitute the value of \( a \)
Now, substitute \( a = 5 \) into the equation:
\[ 3(5^2) + 2d^2 = 93 \]
Calculating \( 5^2 \):
\[ 3(25) + 2d^2 = 93 \]
This simplifies to:
\[ 75 + 2d^2 = 93 \]
Subtract 75 from both sides:
\[ 2d^2 = 93 - 75 \]
\[ 2d^2 = 18 \]
Dividing by 2:
\[ d^2 = 9 \]

### Step 5: Find the value of \( d \)
Taking the square root of both sides:
\[ d = \pm 3 \]
We will take \( d = 3 \) (the positive value).

### Step 6: Find the three numbers
Now we can find the three numbers:
- First number: \( a - d = 5 - 3 = 2 \)
- Second number: \( a = 5 \)
- Third number: \( a + d = 5 + 3 = 8 \)

### Step 7: Identify the greatest number
The three numbers are 2, 5, and 8. The greatest number is:
\[ \text{Greatest number} = 8 \]

### Final Answer
The greatest number among the three is **8**.

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