Consecutive odd numbers (sum of squares = 394): The sum of the squares of two consecutive natural odd numbers is 394. Find the sum of the two numbers.

Difficulty: Easy

Correct Answer: 28

Explanation:

Introduction / Context:
If two consecutive odd numbers are n and n + 2, then their squares sum to a quadratic in n. Solving gives the actual numbers. The final request is the sum of the numbers, which can then be obtained easily once n is known.

Given Data / Assumptions:

Numbers: n and n + 2, both odd, both natural.
n^2 + (n + 2)^2 = 394.

Concept / Approach:
Expand and solve for n via the quadratic formula; choose the positive root. Then compute their sum S = n + (n + 2) = 2n + 2.

Step-by-Step Solution:

n^2 + (n + 2)^2 = 394 ⇒ 2n^2 + 4n + 4 = 394.2n^2 + 4n − 390 = 0 ⇒ divide by 2 ⇒ n^2 + 2n − 195 = 0.Discriminant D = 4 + 780 = 784 ⇒ √D = 28.n = [−2 ± 28]/2 ⇒ n = 13 or n = −15 (reject negative).Numbers: 13 and 15; Sum S = 28.

Verification / Alternative check:
13^2 + 15^2 = 169 + 225 = 394. The sum 28 follows immediately.

Why Other Options Are Wrong:

24, 32, 40: Do not correspond to the valid pair (13, 15).

Common Pitfalls:
Assuming difference 1 rather than 2 for odd consecutive numbers or arithmetic slips in expanding squares.

Consecutive odd numbers (sum of squares = 394): The sum of the squares of two consecutive natural odd numbers is 394. Find the sum of the two numbers.

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