Problem 29 - 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 is 50.

Step 1: Identify the Pattern

Observe the differences between each consecutive pair of numbers. The differences are \(5-2=3\), \(10-5=5\), \(17-10=7\), \(26-17=9\), and \(37-26=11\). The difference between each number is increasing by 2. Hence, the pattern appears to be that each subsequent number is the previous number plus an odd number which increases by 2 with every step.

Step 2: Apply the Pattern

To find the next number in the sequence, add 13 to the last given number. The reason 13 is chosen is because so far, each additional number has been the next odd number in sequence (3, 5, 7, 9, 11). Thus, the next odd number to be added is 13.

Step 3: Calculate the Next Number

The last number given in the sequence is 37. According to our pattern, the next number will be \(37 + 13 = 50\). Therefore, the next number in the sequence should be 50.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are special number patterns where the difference between any two consecutive numbers is always the same. This difference is called the "common difference." In the exercise we looked at, the sequence is not strictly an arithmetic sequence because the differences are not constant; instead, they increase in a regular manner by 2 every step. However, it is important to understand this concept because it forms the basis for many other number patterns.

• An arithmetic sequence example: 3, 7, 11, 15, where the common difference is 4.
• To find the next number, simply add the common difference to the last number in the sequence.
• General formula: If you know the first term and the common difference, the n-th term can be found using: \[a_n = a_1 + (n-1) imes d\] where \(a_1\) is the first term and \(d\) is the common difference.

Understanding arithmetic sequences paves the way to recognizing and solving more complex sequences, like the one in our exercise, which involves an incrementally increasing difference.

Mathematical Reasoning

Mathematical reasoning involves identifying patterns and using logic to arrive at conclusions. In the exercise, you use reasoning to find the pattern in the sequence: the difference between consecutive numbers increases by 2.

• This type of reasoning is essential not only for mathematics but in everyday problem solving.
• When you notice a pattern, consider whether it fits a well-known template like arithmetic sequences, or if it requires deeper analysis.
• Understanding the underlying logic or rule of the pattern helps predict future numbers or solve problems efficiently.

Using this reasoning method enables you to extend your pattern to predict the next number in the sequence, as shown in the solution given.

Problem Solving

Problem solving often involves breaking a problem into manageable parts. With number pattern exercises, this means identifying, understanding, and applying recognized patterns or rules to find a solution.

• The steps in problem solving often include identifying known information, interpreting this information, planning how to proceed, and implementing a solution.
• In situations like our sequence exercise, break down the list to understand the clear relationship between numbers.
• Identify any small changes step by step, like increasing differences, and apply these findings methodically to find the answer.

Problem solving also means validating your solution: checking the work to make sure calculations or conclusions align correctly, confirming your answer is logical and accurate.