Problem 28 - Thinking Mathematically

Question

In Exercises 9-38, identify a pattern in each list of numbers. Then use this pattern to find the next number. (More than one pattern might exist, so it is possible that there is more than one correct answer.) \(2,5,10,17,26,37\),_____

Step-by-Step Solution

Answer

The next number in the sequence: 2, 5, 10, 17, 26, 37 is 48.

Step 1: Observation

Look at the given number sequence carefully: 2, 5, 10, 17, 26, 37. Here, we should try to look for a pattern. Observing, you can see that if you subtract the first element from the second, the result is 3. If you subtract the first element from the third, you get 8. The difference is not constant, so it's not an arithmetic progression. However, look at how these differences are progressing: 3, 8, 15, etc. They're familiar - these are all one less than the squares of positive integers.

Step 2: Identify the Pattern

From the observation in the previous step, we find that the differences are one less than the squares of integers. That is, the numbers in the given sequence are of the form: number in the sequence = square of natural number - 1. Let's check this out: \[2 = 4-2 = 1^2-1, 5 = 4-1 = 2^2-1, 10 = 9-1 = 3^2-1, 17 = 16-1 = 4^2-1, 26 = 25-1 = 5^2-1, 37 = 36-1 = 6^2-1\] This confirms the pattern.

Step 3: Find the Next Number

To find the next number in the sequence, we would simply extend the pattern. The next number will be \(7^2-1 = 49-1\).

Step 4: Result

After performing the operations above, we find that the next number in the sequence would be 48.