In an Arithmetic Progression (AP), 'l' (lowercase L) commonly refers to the last term of the sequence. While the question uses 'L' (uppercase), standard mathematical notation for the last term in an AP is 'l'. This term represents the final value in a finite arithmetic sequence.
An Arithmetic Progression is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference (d).
Understanding the Last Term (l) in an AP
The last term 'l' is a crucial component when analyzing or calculating properties of a finite arithmetic progression. It can also be denoted as an, representing the n-th term of the sequence.
To find the last term 'l' (or the n-th term an) of an arithmetic progression, the following formula is used:
l = a + (n - 1)d
Where:
a: The first term of the arithmetic progression.n: The total number of terms in the sequence.d: The common difference between consecutive terms.
Key Components of an Arithmetic Progression
Understanding each variable in the AP formula is essential:
- First Term (a): This is the starting number of the sequence. For example, in the AP
2, 5, 8, 11, the first termais 2. - Common Difference (d): This is the constant value added to each term to get the next term. It can be found by subtracting any term from its succeeding term. In
2, 5, 8, 11, the common differencedis5 - 2 = 3. - Number of Terms (n): This indicates how many numbers are in the sequence. In
2, 5, 8, 11, the number of termsnis 4. - Last Term (l or an): This is the final number in the sequence. In
2, 5, 8, 11, the last termlis 11.
Variable Summary
| Variable | Description | Example (from 2, 5, 8, 11) |
|---|---|---|
| a | The first term | 2 |
| d | The common difference | 3 |
| n | The number of terms | 4 |
| l | The last term (or an, the n-th term) | 11 |
For a deeper dive into arithmetic progressions, you can refer to resources like Wikipedia's page on Arithmetic Progressions.
Practical Applications and Examples
The formula l = a + (n - 1)d is versatile and can be rearranged to find any of the variables if the others are known.
Example: Finding the First Term when the Last Term is Known
Consider a scenario where you know the last term, common difference, and number of terms, and you need to find the first term. For instance, if the last term (l) is 20, the common difference (d) is -1, and the number of terms (n) is 17, we can find the first term (a) using the formula:
l = a + (n - 1)d
Substitute the given values into the formula:
20 = a + (17 - 1)(-1)
First, simplify the expression within the parentheses:
20 = a + (16)(-1)
Next, perform the multiplication:
20 = a - 16
Finally, isolate 'a' by adding 16 to both sides of the equation:
a = 20 + 16
a = 36
Thus, the first term of this arithmetic progression would be 36.
Other Common Uses:
- Finding the Last Term: If you have
a,n, andd, you can directly calculatel. - Finding the Number of Terms: If
a,l, anddare known, you can rearrange the formula to findn.n = ((l - a) / d) + 1
- Finding the Common Difference: If
a,l, andnare known, you can rearrange to findd.d = (l - a) / (n - 1)
Understanding the concept of 'l' as the last term of an AP is fundamental for solving various problems related to sequences and series.
