1. Definition of an Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence. Instead of listing the numbers, we add them together to find a total. This total is often written as \( S_n \), meaning the sum of the first \( n \) terms.
Example:
Sequence: \( 3, 6, 9, 12 \)
Series: \( 3 + 6 + 9 + 12 = 30 \)
2. Sum of First \(n\) Terms of an AP
The sum of the first \( n \) terms of an arithmetic sequence can be found in different ways. Two popular methods are the pairing method and using the nth-term formula.
2.1. Using the Pairing Method
The pairing method adds terms from the beginning and end of the sequence. Each pair has the same total.
Example: \( 2 + 5 + 8 + 11 + 14 \)
- 1st + last = \( 2 + 14 = 16 \)
- 2nd + 4th = \( 5 + 11 = 16 \)
Each pair adds up to 16. If there are \( k \) such pairs, then:
\( S_n = \text{number of pairs} \times \text{sum of each pair} \)
2.2. Using the nth Term Formula
We can also find the sum using the formula for the nth term:
\( a_n = a + (n - 1)d \)
Using this, the sum becomes:
\( S_n = \dfrac{n}{2}(a + a_n) \)
This method is helpful when the last term is known or easy to find.
3. Formulas for \(S_n\)
There are two main formulas used to find the sum of the first \( n \) terms of an arithmetic sequence. Both give the same result, and the choice depends on which information is available.
3.1. Formula \(S_n = \dfrac{n}{2}(a + l)\)
Here:
- \( a \) = first term
- \( l \) = last term (nth term)
The formula is:
\( S_n = \dfrac{n}{2}(a + l) \)
This works best when the first and last terms are known.
3.2. Formula \(S_n = \dfrac{n}{2}[2a + (n-1)d]\)
When the last term is not given, we use this expanded version:
\( S_n = \dfrac{n}{2}[2a + (n - 1)d] \)
This formula uses the first term and the common difference, so it works even without knowing the final term.
4. Applications of AP Sum
The formulas for arithmetic series are useful for finding missing information such as the number of terms or the common difference.
4.1. Finding Number of Terms
If you know the sum and the first and last terms, you can rearrange the formula to find how many terms are present.
Example:
If \( S = 75 \), \( a = 5 \), and \( l = 25 \):
\( 75 = \dfrac{n}{2}(5 + 25) \Rightarrow 75 = 15n \Rightarrow n = 5 \)
4.2. Finding Common Difference or First Term
Using the sum formulas, you can solve for \( d \) or \( a \) by substituting known values and solving the resulting equation.
Example:
Given: \( S_6 = 54 \), \( a = 3 \)
\( 54 = \dfrac{6}{2}[2(3) + 5d] \)
\( 54 = 3(6 + 5d) \)
\( 18 = 6 + 5d \Rightarrow d = \dfrac{12}{5} \)
5. Examples
These examples show how arithmetic series appear in simple number patterns and everyday situations.
5.1. Simple AP Sum Examples
- Sum of first 10 terms of \( 4, 7, 10, 13, \ldots \)
- Sum of first 5 terms of \( 20, 16, 12, 8, \ldots \)
- Sum of first 8 terms of \( -1, 2, 5, 8, \ldots \)
Each pattern can be solved using either pairing or formula methods.
5.2. Real-Life Applications of AP Series
Arithmetic series show up in situations involving steady growth or decline. Examples include:
- Total distance covered when speed increases steadily
- Total savings when adding a fixed amount regularly
- Total seats in rows where each row has a fixed number more than the previous one
These situations follow the same idea of adding terms with a constant difference.