The sum of three numbers in A.P. is 15 whereas sum of their squares is 83. The numbers are……………………..

Answer

To solve the problem, we need to find three numbers in Arithmetic Progression (A.P.) such that their sum is 15 and the sum of their squares is 83. Let's denote the three numbers as \( a - d \), \( a \), and \( a + d \), where \( a \) is the middle term and \( d \) is the common difference.

### Step-by-Step Solution:

1. **Set Up the Equations:**
The sum of the three numbers can be expressed as:
\[ (a - d) + a + (a + d) = 15 \]
Simplifying this gives:
\[ 3a = 15 \]
Therefore:
\[ a = 5 \]

**Hint:** Use the property of A.P. that the sum of the terms can be expressed in terms of the middle term.

2. **Sum of Squares:**
Next, we need to set up the equation for the sum of their squares:
\[ (a - d)^2 + a^2 + (a + d)^2 = 83 \]
Substituting \( a = 5 \) into the equation:
\[ (5 - d)^2 + 5^2 + (5 + d)^2 = 83 \]

**Hint:** Remember to expand the squares correctly.

3. **Expand the Squares:**
Expanding the squares gives:
\[ (25 - 10d + d^2) + 25 + (25 + 10d + d^2) = 83 \]
Combining like terms results in:
\[ 25 - 10d + d^2 + 25 + 25 + 10d + d^2 = 83 \]
This simplifies to:
\[ 75 + 2d^2 = 83 \]

**Hint:** Combine the constant terms and the \( d^2 \) terms carefully.

4. **Solve for \( d^2 \):**
Rearranging the equation gives:
\[ 2d^2 = 83 - 75 \]
Thus:
\[ 2d^2 = 8 \]
Dividing both sides by 2:
\[ d^2 = 4 \]
Taking the square root gives:
\[ d = 2 \quad \text{or} \quad d = -2 \]

**Hint:** Remember that \( d \) can be both positive and negative in A.P.

5. **Find the Numbers:**
Now, substituting \( d \) back to find the numbers:
- If \( d = 2 \):
\[ a - d = 5 - 2 = 3, \quad a = 5, \quad a + d = 5 + 2 = 7 \]
So the numbers are \( 3, 5, 7 \).

- If \( d = -2 \):
\[ a - d = 5 - (-2) = 7, \quad a = 5, \quad a + d = 5 + (-2) = 3 \]
So the numbers are \( 7, 5, 3 \).

**Hint:** Check both cases for \( d \) to ensure all possible combinations are found.

### Final Answer:
The numbers are \( 3, 5, 7 \) or \( 7, 5, 3 \).