Three numbers are in arithmetic progression with a total sum of 30 and their product equal to 910. What is the greatest of these three numbers?

Introduction / Context:
This question combines properties of arithmetic progressions with algebraic manipulation involving sums and products. You are given that three numbers are in arithmetic progression, their sum is known, and their product is also known. You need to determine the largest of the three numbers. This type of problem checks your ability to model numbers in an arithmetic progression and to use algebra to simplify and solve the resulting equations.

Given Data / Assumptions:

• Three numbers are in arithmetic progression.
• The sum of the three numbers is 30.
• The product of the three numbers is 910.
• We are asked to find the greatest of the three numbers.
• All three numbers are real numbers and the options suggest integer solutions.

Concept / Approach:
When three numbers are in arithmetic progression, they can be written as a - d, a, and a + d, where a is the middle term and d is the common difference. The sum of these three numbers is (a - d) + a + (a + d) = 3a, so we can quickly find the middle term from the sum. Then we use the product condition to determine d. Once we know a and d, we can write down all three numbers and identify the greatest. This method neatly uses the structure of an arithmetic progression.

Step-by-Step Solution:
Let the three numbers be a - d, a, and a + d. Their sum is (a - d) + a + (a + d) = 3a. Given that the sum is 30, we have 3a = 30, so a = 30 / 3 = 10. Therefore the three numbers are 10 - d, 10, and 10 + d. Their product is (10 - d) * 10 * (10 + d) = 910. Notice that (10 - d) * (10 + d) = 100 - d^2 by the difference of squares formula. So 10 * (100 - d^2) = 910. Divide both sides by 10 to simplify: 100 - d^2 = 91. Rearrange to find d^2: d^2 = 100 - 91 = 9. So d = 3 or d = -3, but in either case the three numbers are 7, 10, and 13. The greatest of these three numbers is 13.

Verification / Alternative Check:
We can verify by checking the sum and product directly. For the numbers 7, 10, and 13, the sum is 7 + 10 + 13 = 30, which matches the given sum. The product is 7 * 10 * 13 = 70 * 13 = 910, which matches the given product. The three numbers are equally spaced by 3, confirming they form an arithmetic progression. Since 13 is clearly larger than 7 and 10, our identification of the greatest number is correct.

Why Other Options Are Wrong:

• Option 17: If 17 were the greatest, numbers like 13, 15, 17 might be considered, but their sum is 45, not 30, and their product is not 910.
• Option 15: For 9, 12, 15 the sum is 36, not 30, and the product is not 910 either.
• Option 10: This is the middle term, not the greatest term, in the correct arithmetic progression.
• Option 19: Any triplet with 19 as the largest term and in arithmetic progression would lead to a different sum and product, so it cannot satisfy the given conditions.

Common Pitfalls:
One common mistake is to assume the three numbers are a, a + d, and a + 2d instead of centering them as a - d, a, a + d. That approach can also work but leads to different algebra and sometimes more complex equations. Another error is to misapply the difference of squares formula or to mishandle the product condition. Students also sometimes forget that d can be positive or negative, but in this question both possibilities give the same set of three numbers, just in reversed order. Being systematic about the algebra avoids confusion.

Final Answer:
The greatest of the three numbers in arithmetic progression is 13.