Arithmetic Progression Problem Solving Determining Terms In A Sequence
In the realm of mathematics, arithmetic progressions (APs) hold a significant position. They represent sequences of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference, often denoted by 'd'. Understanding arithmetic progressions is crucial for various mathematical applications, including pattern recognition, financial calculations, and computer science algorithms. This article delves into the intricacies of arithmetic progressions, focusing on determining terms and verifying if a given number belongs to a specific AP. We will explore the fundamental concepts, formulas, and step-by-step procedures involved in solving such problems. Specifically, we will address the question: Given that the 5th term of an AP is 28 and the 9th term is 48, are 63 and 78 terms of this AP?
Understanding Arithmetic Progressions
To tackle the problem at hand, a solid understanding of arithmetic progressions is essential. An arithmetic progression is characterized by a sequence of numbers where the difference between consecutive terms is constant. This constant difference, denoted as 'd', is the cornerstone of APs. The first term of an AP is typically represented by 'a'. With these two parameters, 'a' and 'd', we can define any term in the sequence. The nth term of an AP, denoted as a_n, can be calculated using the formula:
a_n = a + (n - 1)d
where:
- a_n is the nth term
- a is the first term
- n is the term number
- d is the common difference
This formula is fundamental in solving problems related to arithmetic progressions. It allows us to find any term in the sequence if we know the first term and the common difference. Conversely, if we know two terms of the AP, we can determine both the first term and the common difference. This is precisely the scenario presented in our problem, where we are given the 5th and 9th terms.
Solving for the First Term and Common Difference
In our problem, we are given that the 5th term (a_5) is 28 and the 9th term (a_9) is 48. Using the formula for the nth term, we can set up two equations:
- a_5 = a + 4d = 28
- a_9 = a + 8d = 48
Now we have a system of two linear equations with two unknowns (a and d). We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method. Subtracting equation (1) from equation (2), we get:
(a + 8d) - (a + 4d) = 48 - 28
This simplifies to:
4d = 20
Dividing both sides by 4, we find the common difference:
d = 5
Now that we have the common difference, we can substitute it back into either equation (1) or (2) to solve for the first term (a). Let's substitute d = 5 into equation (1):
a + 4(5) = 28
a + 20 = 28
Subtracting 20 from both sides, we get:
a = 8
Therefore, the first term of the AP is 8, and the common difference is 5. Now we have a complete description of the AP, allowing us to determine any term in the sequence.
Verifying Membership of 63 and 78 in the AP
The next part of our problem asks whether 63 and 78 are terms of this AP. To determine this, we need to see if there exists an integer value of 'n' for which the nth term (a_n) equals 63 or 78. We will use the formula for the nth term again:
a_n = a + (n - 1)d
We know that a = 8 and d = 5. Let's first check for 63:
63 = 8 + (n - 1)5
Subtracting 8 from both sides, we get:
55 = (n - 1)5
Dividing both sides by 5, we get:
11 = n - 1
Adding 1 to both sides, we find:
n = 12
Since n = 12 is an integer, 63 is a term of the AP. Specifically, it is the 12th term. Now, let's check for 78:
78 = 8 + (n - 1)5
Subtracting 8 from both sides, we get:
70 = (n - 1)5
Dividing both sides by 5, we get:
14 = n - 1
Adding 1 to both sides, we find:
n = 15
Since n = 15 is also an integer, 78 is a term of the AP as well. It is the 15th term.
Conclusion
In this article, we've explored the concept of arithmetic progressions and demonstrated how to solve problems involving finding terms and verifying membership. We successfully determined the first term and common difference of an AP given two terms, and then used this information to verify whether 63 and 78 are terms of the AP. We found that both 63 and 78 are indeed terms of the AP, specifically the 12th and 15th terms, respectively. Understanding and applying the formula for the nth term of an AP is crucial for solving a wide range of mathematical problems. This knowledge is not only valuable in academic settings but also in practical applications where sequences and patterns are involved. The ability to analyze and manipulate arithmetic progressions provides a solid foundation for further exploration of mathematical concepts and problem-solving techniques. This exploration highlights the importance of arithmetic progressions in mathematics and demonstrates how to effectively solve problems related to them. By understanding the fundamental concepts and applying the appropriate formulas, we can confidently tackle various challenges involving arithmetic sequences.
Arithmetic Progression, nth term, common difference, first term, sequence, mathematical problems
