Unraveling the Mystery of Arithmetic Progressions: A Step by Step Guide with an Example
Arithmetic progressions are a fundamental concept in mathematics, often encountered in various fields such as engineering, finance, and computer science. This article will delve into the details of arithmetic progressions, elucidating the formula for the nth term (Tn) and demonstrating a practical example.
Understanding Arithmetic Progressions
An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference (d), to the preceding term. The formula for the nth term of an arithmetic progression is given by:
Tn an (n - 1)d
Where:
Tn is the nth term of the sequence a1 is the first term of the sequence d is the common difference n is the term numberCase Study: A Real World Example
Consider the arithmetic progression: 9, -2, 5, ...
Upon closer inspection, we notice that the first term (a1) is 9, which is incorrect based on the given terms. The correct first term should be -9. This incorrect term can be fixed by identifying the common difference and then recalculating the sequence.
Identifying the Common Difference (d)
To find the common difference (d), we subtract the first term from the second term:
d a2 - a1
Given the corrected first term (a1 -9) and the second term (a2 -2), we calculate the common difference as follows:
d -2 - (-9) -2 9 7
Finding the nth Term (Tn) of the Sequence
Given the value of Tn 131, we proceed to find the term number (n). We use the Tn formula and substitute the known values:
Tn an (n - 1)d
131 -9(n - 1) 7n
131 -9n 9 7n
131 -2n 9
131 - 9 -2n
122 -2n
n 122 / -2 -61
Upon re-evaluating the steps, we notice an error. The correct calculation should be:
131 -9 (n - 1) * 7
131 -9 7n - 7
131 7n - 16
131 16 7n
147 7n
n 147 / 7 21
Therefore, the term number (n) is 21.
Conclusion
Understanding arithmetic progressions and the nth term formula is essential for addressing various mathematical and real-world problems. This example demonstrates the importance of correctly identifying the first term and common difference and applying the correct formula to solve for the term number (n).
Frequently Asked Questions (FAQs)
What is an arithmetic progression?
An arithmetic progression is a sequence of numbers where each term after the first is obtained by adding a constant value (common difference) to the preceding term.
How do you find the nth term of an arithmetic progression?
Using the formula Tn an (n - 1)d, where Tn is the nth term, an is the first term, d is the common difference, and n is the term number.
What is the common difference in an arithmetic progression?
The common difference (d) is the constant value added to each term to obtain the next term in the sequence.