Split a number into three AP parts: The number 15 is divided into three parts in AP, and the sum of their squares is 83. What is the smallest part?

Introduction / Context:
When three numbers are in AP, they can be written as (a − d), a, (a + d). Their sum and sum of squares yield two equations in a and d. Solving provides the parts and identifies the smallest one.

Given Data / Assumptions:

• Sum = 15 ⇒ (a − d) + a + (a + d) = 3a = 15.
• Sum of squares = 83 ⇒ (a − d)^2 + a^2 + (a + d)^2 = 83.

Concept / Approach:
Use 3a = 15 ⇒ a = 5. For the squares, expand and simplify to 3a^2 + 2d^2 = 83, then solve for d. Smallest part is a − d.

Step-by-Step Solution:

Verification / Alternative check:
Check squares: 3^2 + 5^2 + 7^2 = 9 + 25 + 49 = 83. Perfect match.

Why Other Options Are Wrong:

• 5, 6, 8: Not the smallest member of the valid triplet 3, 5, 7.

Common Pitfalls:
Dropping the 2 in 2d^2 when summing squares, or overlooking that d can be ±2 (we need the smallest, thus a − d).

Final Answer: