An Arrangement Of Billiard Balls Has 5 Balls In The First Row And 16 Rows In All. In Each Successive Row, One Ball Is Added.What Is The Explicit Rule For This Situation, And How Many Balls Will Be In The 16th Row?Drag And Drop The Answers Into The
An Arrangement of Billiard Balls: Understanding the Explicit Rule and Calculating the Number of Balls in the 16th Row
The arrangement of billiard balls is a classic problem in mathematics that involves understanding patterns and sequences. In this scenario, we have a total of 16 rows, with 5 balls in the first row and one additional ball in each successive row. The problem requires us to identify the explicit rule governing this situation and determine the number of balls in the 16th row.
To begin, let's examine the pattern of the number of balls in each row. The first row has 5 balls, the second row has 6 balls, the third row has 7 balls, and so on. This pattern suggests that the number of balls in each row is increasing by 1.
The explicit rule governing this situation can be expressed as a simple arithmetic sequence. In an arithmetic sequence, each term is obtained by adding a fixed constant to the previous term. In this case, the constant is 1, and the first term is 5.
The explicit rule can be written as:
a_n = 5 + (n - 1) * 1
where a_n is the number of balls in the nth row.
We can simplify the explicit rule by evaluating the expression:
a_n = 5 + (n - 1)
This simplification allows us to easily calculate the number of balls in any row.
To determine the number of balls in the 16th row, we can substitute n = 16 into the simplified explicit rule:
a_16 = 5 + (16 - 1) a_16 = 5 + 15 a_16 = 20
Therefore, the number of balls in the 16th row is 20.
In conclusion, the explicit rule governing the arrangement of billiard balls is an arithmetic sequence with a first term of 5 and a common difference of 1. By applying this rule, we can easily calculate the number of balls in any row. In this case, we found that the number of balls in the 16th row is 20.
- Mathematical Concepts: This problem involves understanding arithmetic sequences and explicit rules.
- Problem-Solving Strategies: The problem requires the application of mathematical concepts to solve a real-world scenario.
- Critical Thinking: The problem requires critical thinking to identify the explicit rule and calculate the number of balls in the 16th row.
- Arithmetic Sequences: This problem involves understanding arithmetic sequences and their applications.
- Explicit Rules: The problem requires the identification of an explicit rule governing a situation.
- Pattern Recognition: The problem involves recognizing patterns in a sequence of numbers.
- Introduction to Arithmetic Sequences: This article provides an introduction to arithmetic sequences and their applications.
- Explicit Rules in Mathematics: This article discusses the concept of explicit rules in mathematics and their applications.
- Pattern Recognition in Mathematics: This article provides an overview of pattern recognition in mathematics and its applications.
Q&A: Understanding the Arrangement of Billiard Balls
In our previous article, we explored the arrangement of billiard balls and identified the explicit rule governing this situation. We also calculated the number of balls in the 16th row. In this article, we will answer some frequently asked questions related to this topic.
A: The explicit rule for the arrangement of billiard balls is an arithmetic sequence with a first term of 5 and a common difference of 1. This means that each term in the sequence is obtained by adding 1 to the previous term.
A: To calculate the number of balls in any row, you can use the simplified explicit rule:
a_n = 5 + (n - 1)
where a_n is the number of balls in the nth row.
A: You can use the same explicit rule to find the number of balls in any row. For example, to find the number of balls in the 10th row, you would substitute n = 10 into the explicit rule:
a_10 = 5 + (10 - 1) a_10 = 5 + 9 a_10 = 14
A: No, the explicit rule is only applicable to positive integers. If you try to find the number of balls in a row that is not a positive integer, you will get an invalid result.
A: The explicit rule for the arrangement of billiard balls is related to other mathematical concepts such as arithmetic sequences and pattern recognition. It is also a simple example of a linear function.
A: Yes, the explicit rule for the arrangement of billiard balls can be used to solve other problems that involve arithmetic sequences and pattern recognition. However, you will need to adapt the rule to fit the specific problem you are trying to solve.
A: The explicit rule for the arrangement of billiard balls has several real-world applications, including:
- Finance: The rule can be used to calculate the number of periods in a financial investment or loan.
- Science: The rule can be used to model the growth of a population or the spread of a disease.
- Engineering: The rule can be used to design and optimize systems that involve linear growth or decay.
In conclusion, the explicit rule for the arrangement of billiard balls is a simple and powerful tool that can be used to solve a wide range of problems. By understanding this rule and its applications, you can develop a deeper appreciation for the beauty and power of mathematics.
- Mathematical Concepts: This article involves understanding arithmetic sequences, pattern recognition, and linear functions.
- Problem-Solving Strategies: The article requires the application of mathematical concepts to solve real-world problems.
- Critical Thinking: The article requires critical thinking to identify the explicit rule and its applications.
- Arithmetic Sequences: This article provides an introduction to arithmetic sequences and their applications.
- Explicit Rules in Mathematics: This article discusses the concept of explicit rules in mathematics and their applications.
- Pattern Recognition in Mathematics: This article provides an overview of pattern recognition in mathematics and its applications.
- Introduction to Arithmetic Sequences: This article provides an introduction to arithmetic sequences and their applications.
- Explicit Rules in Mathematics: This article discusses the concept of explicit rules in mathematics and their applications.
- Pattern Recognition in Mathematics: This article provides an overview of pattern recognition in mathematics and its applications.