Problem 35 - 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.) \(64,-16,4,-1\),_____

Step-by-Step Solution

Answer

The next number in the sequence is '-i'.

Step 1: Compare Adjacent Numbers

Start by looking at how each number in the list relates to the one before it. Draw some conclusions from your observations. The numbers sequence is \(64, -16, 4, -1\). When observing these numbers, it can be seen that each number is a multiple of the previous number with alternate signs. Further, observe that each number, an absolute value, is the square root of the preceding number.

Step 2: Identify Pattern of the Sequence

Based on the observation above, a pattern can be noticed. The pattern shows that each subsequent number is the square root of the previous one and the sign alternates. Mathematically, this can be represented as \((-1)^n \sqrt {x}\), where n is the position of the number in the sequence and x is the previous number.

Step 3: Apply the Pattern to Find Next Number

Apply the formula \((-1)^n \sqrt {x}\) on the last number to find the next number in the sequence. Here \(n=5\) (as it is the 5th position), and \(x=-1\). Using this we get \( = (-1)^5 \cdot \sqrt{-1} = -i.\) Here i represents the imaginary unit, which equals to the square root of -1. Therefore, in this context '-i' is a valid response, even though it might be unusual in this context.

Key Concepts

Number Sequences

Number sequences involve a list of numbers that follow a certain pattern or rule. Understanding these patterns allows you to predict the next number in the sequence. For example, in the sequence given:

• 64
• -16
• 4
• -1

You can observe the numbers alternating in sign while decreasing in absolute value, indicating a possible mathematical rule. Patterns can relate to operations like addition, subtraction, multiplication, or division. They can also involve more complex rules such as taking square roots, as seen in this sequence.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In many sequences, recognizing when to use square roots can help unlock the pattern. For the sequence 64, -16, 4, -1, each number is the square root of the previous number:

• The square root of 64 is 8. Another step reveals:
• The square root of 16 is 4 (with a sign change to negative 16)
• The square root of 4 is 2, switching sign to -1

Understanding square roots provides insight into how these numbers connect.

Imaginary Numbers

Imaginary numbers arise when you need to find the square root of a negative number. The number 'i' represents the square root of -1, providing a way to work with otherwise unsolvable roots.In the sequence above, recognizing imaginary numbers is critical for continuing the pattern after -1. Applying the formula \((-1)^5 \cdot \sqrt{-1} = -i\)reveals that the next number in this sequence is

• -i

Understanding and identifying when to use imaginary numbers allows for expansion into new realms of mathematics.

Alternating Signs

Alternating signs in a sequence indicate a change of direction in the positive or negative nature of numbers. An alternating sign sequence can be expressed as \((-1)^n\).In the sequence 64, -16, 4, -1, the signs alternate with each step:

• 64 (positive)
• -16 (negative)
• 4 (positive)
• -1 (negative)

This sign pattern plays a critical role in applying the formula correctly to find the next number. Identifying alternating signs helps confirm and guide the sequence's progression.