A number 15 is divided into three parts which are in AP and the sum of their squares is 83. Find the smallest number.

Options

A. 5
B. 3
C. 6
D. 8

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the Parts
Let the three parts be:
- First part: \( a - d \)
- Second part: \( a \)
- Third part: \( a + d \)

Since these parts are in Arithmetic Progression (AP), the middle part is \( a \).

### Step 2: Set Up the Equation for the Sum
According to the problem, the sum of these three parts is 15:
\[ (a - d) + a + (a + d) = 15 \]
This simplifies to:
\[ 3a = 15 \]
From this, we can solve for \( a \):
\[ a = \frac{15}{3} = 5 \]

### Step 3: Set Up the Equation for the Sum of Squares
The sum of the squares of these parts is given as 83:
\[ (a - d)^2 + a^2 + (a + d)^2 = 83 \]
Substituting \( a = 5 \):
\[ (5 - d)^2 + 5^2 + (5 + d)^2 = 83 \]

### Step 4: Expand the Squares
Expanding each term:
\[ (5 - d)^2 = 25 - 10d + d^2 \]
\[ 5^2 = 25 \]
\[ (5 + d)^2 = 25 + 10d + d^2 \]
Now, substituting these back into the equation:
\[ (25 - 10d + d^2) + 25 + (25 + 10d + d^2) = 83 \]

### Step 5: Combine Like Terms
Combining all the terms gives:
\[ 25 - 10d + d^2 + 25 + 25 + 10d + d^2 = 83 \]
This simplifies to:
\[ 75 + 2d^2 = 83 \]

### Step 6: Solve for \( d^2 \)
Now, isolate \( d^2 \):
\[ 2d^2 = 83 - 75 \]
\[ 2d^2 = 8 \]
\[ d^2 = 4 \]
Taking the square root gives:
\[ d = \pm 2 \]

### Step 7: Find the Three Parts
Now we can find the three parts:
1. If \( d = 2 \):
- First part: \( 5 - 2 = 3 \)
- Second part: \( 5 \)
- Third part: \( 5 + 2 = 7 \)

2. If \( d = -2 \):
- First part: \( 5 - (-2) = 7 \)
- Second part: \( 5 \)
- Third part: \( 5 + (-2) = 3 \)

### Conclusion
In both cases, the smallest number is 3.

### Final Answer
The smallest number is **3**.