Unraveling the Pattern: Finding the nth Term of an Arithmetic Progression
Have you ever encountered a sequence of numbers where the difference between consecutive terms is always the same? That, my friends, is an Arithmetic Progression (AP). These sequences are all around us, from the pricing of certain subscription services to the way some sports statistics are presented. Today, we’re going to tackle a specific problem: finding the general formula for the nth term of an AP when we know two specific terms. Specifically, we'll be answering the question: What is the nth term of an AP whose 6th term is 12 and 8th term is 22?
Understanding Arithmetic Progressions (APs)
Before we dive into the solution, let's quickly recap what an AP is. An AP is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, usually denoted by the letter 'd'.
The general formula for the nth term of an AP is given by:
an = a1 + (n - 1)d
Where:
- an represents the nth term of the AP.
- a1 represents the first term of the AP.
- n represents the position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.).
- d represents the common difference.
The Problem at Hand
We are given the following information:
- The 6th term (a6) is 12.
- The 8th term (a8) is 22.
Our goal is to find an, the general formula for any term in this specific AP.
Step 1: Finding the Common Difference (d)
Since we know two terms and their positions, we can use them to find the common difference. The difference between any two terms in an AP is equal to the product of the common difference and the difference in their positions. In other words, the difference between the 8th term and the 6th term is equivalent to the common difference multiplied by (8 - 6) positions.
We can express this mathematically:
a8 - a6 = (8 - 6)d
Now, let's plug in the given values:
22 - 12 = (2)d
10 = 2d
To find 'd', we simply divide both sides by 2:
d = 10 / 2
d = 5
So, the common difference of this AP is 5. This means that each term increases by 5 as we move along the sequence.
Step 2: Finding the First Term (a1)
Now that we have the common difference (d = 5), we can use either of the given terms to find the first term (a1). Let's use the 6th term (a6 = 12) and the general formula for the nth term:
an = a1 + (n - 1)d
Substituting the values for the 6th term (n = 6, a6 = 12, and d = 5):
12 = a1 + (6 - 1) * 5
12 = a1 + (5) * 5
12 = a1 + 25
To isolate a1, subtract 25 from both sides:
a1 = 12 - 25
a1 = -13
So, the first term of this arithmetic progression is -13.
Step 3: Deriving the nth Term Formula (an)
We now have all the components needed to write the general formula for the nth term of this AP:
- First term (a1) = -13
- Common difference (d) = 5
Using the general formula an = a1 + (n - 1)d, we substitute our values:
an = -13 + (n - 1) * 5
Now, we simplify the expression:
an = -13 + 5n - 5
an = 5n - 13 - 5
an = 5n - 18
Conclusion
Therefore, the nth term of an AP whose 6th term is 12 and 8th term is 22 is given by the formula an = 5n - 18.
Let's quickly check our work. If n = 6:
a6 = 5(6) - 18 = 30 - 18 = 12. (Correct!)
If n = 8:
a8 = 5(8) - 18 = 40 - 18 = 22. (Correct!)
This formula allows us to find any term in this specific arithmetic progression just by plugging in its position 'n'.
Frequently Asked Questions (FAQ)
Q1: How do I find the common difference if I'm given two terms in an AP?
To find the common difference ('d') when you know two terms (say, the mth term am and the nth term an), you can use the formula: d = (an - am) / (n - m). This essentially calculates the total difference between the terms and divides it by the number of steps (differences) between them.
Q2: Why is the formula for the nth term important in an AP?
The nth term formula (an = a1 + (n - 1)d) is crucial because it provides a shortcut to find any term in an arithmetic progression without having to list out all the preceding terms. Once you know the first term (a1) and the common difference (d), you can instantly calculate the value of any term by simply substituting its position ('n') into the formula.
Q3: What if the terms given are not consecutive?
The method shown in this article is specifically designed to work even when the given terms are not consecutive. The key is to use the difference in the term positions (n - m) when calculating the common difference. This accounts for the number of steps between those non-consecutive terms.
Q4: How can I verify my calculated first term (a1)?
After calculating 'd' and 'a1', you can verify your first term by using one of the original given terms. For example, if you found 'd' and 'a1', plug them back into the nth term formula for the position of one of the given terms. If the result matches the given term's value, your 'a1' is likely correct.
