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Chapter 1: Problem 16

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.) \(1,5,25,125\), ____

Short Answer

Expert verified
625

Step by step solution

01

Identify the Pattern

Look at the list \(1, 5, 25, 125\). Notice that each following number in the list is a multiple of its predecessor. The first number is multiplied by 5 to get the second number \(1*5=5\), the second number is multiplied by 5 to get the third number \(5*5=25\), and the third number is multiplied by 5 to get the fourth number \(25*5=125\). Thus, the pattern is that each number in the list is 5 times the previous number.
02

Apply the Pattern

Apply the identified pattern to the last number in the list. This means multiply the last number \(125\) by 5.
03

Calculate the Next Number

The calculation is \(125*5=625\). Therefore, the next number in the list, following the identified pattern, is 625.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Patterns in Mathematics
Identifying patterns in mathematics can be both intuitive and logical. It involves looking at sequences and trying to find relationships between numbers. This type of reasoning can help to predict future numbers in a sequence. Patterns can vary greatly. They may involve addition, subtraction, multiplication, or even more complex mathematical operations. For the sequence given in the exercise, we can see a clear multiplication pattern. Understanding these patterns helps to build a strong foundation in mathematical thinking.
  • Types of Patterns: These may include reoccurring sequences, growth, and decay patterns.
  • Visual patterns: Sometimes drawing or visualizing numbers helps in recognizing patterns.
Patterns help in simplifying complex problems and help sharpen problem-solving skills.
Multiplication Sequences
In the exercise, the number series falls under the category of multiplication sequences. A multiplication sequence is a list of numbers where each number is a product of the previous number and a fixed multiplier. In this case, each subsequent number in the sequence is derived by multiplying the previous number by 5. Multiplication sequences are straightforward once the multiplier is identified. They are predictable and are an essential part of understanding mathematical progressions.
  • The given series: 1, 5, 25, 125 is created by multiplying the previous number by 5.
  • Multiplier: This is the constant value used to get the next number. Here the multiplier is 5.
Such sequences are often used in algebra and help in understanding exponential growth.
Arithmetic Reasoning
Arithmetic reasoning involves analyzing numbers to identify the logic or pattern behind an arrangement. It鈥檚 akin to solving a puzzle by looking at clues, where numbers give hints about the sequence. Recognizing the pattern in the original exercise requires analyzing how each number is related to the one before it. Given an arithmetic problem, logical steps can simplify the complexity.
  • The rule here was simple: multiply by 5 to get the next number.
  • To solve, apply the identified rule consistently to the last known number in the sequence.
This logic is fundamental in solving various mathematical problems and is a skill that enhances as one practices more arithmetic problems.

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