Arithmetic Progression (AP)
Arithmetic Progression (AP)
Arithmetic Progression (AP) is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is referred to as the common difference of the AP. Arithmetic progressions are among the simplest and most commonly used sequences in mathematics.
Understanding Arithmetic Progression
An arithmetic progression is a sequence of the form:
[ a, a + d, a + 2d, a + 3d, \ldots ]
where:
- ( a ) is the first term,
- ( d ) is the common difference between the terms.
Important Points and Differences
| Aspect | Description |
|---|---|
| First Term (( a )) | The starting number of the sequence. |
| Common Difference (( d )) | The constant amount that each term increases or decreases by. |
| ( n )-th Term | The term at the ( n )-th position in the sequence. |
| Sum of ( n ) Terms | The total of the first ( n ) terms in the sequence. |
Formulas in Arithmetic Progression
( n )-th Term of an AP
The ( n )-th term of an arithmetic progression can be found using the formula:
[ a_n = a + (n - 1)d ]
where:
- ( a_n ) is the ( n )-th term,
- ( a ) is the first term,
- ( n ) is the term number,
- ( d ) is the common difference.
Sum of the First ( n ) Terms of an AP
The sum of the first ( n ) terms of an arithmetic progression can be calculated using the formula:
[ S_n = \frac{n}{2} [2a + (n - 1)d] ]
or equivalently:
[ S_n = \frac{n}{2} (a + a_n) ]
where:
- ( S_n ) is the sum of the first ( n ) terms,
- ( a ) is the first term,
- ( a_n ) is the ( n )-th term,
- ( n ) is the number of terms to be added.
Examples to Explain Important Points
Example 1: Finding the ( n )-th Term
Consider an arithmetic progression where the first term ( a = 5 ) and the common difference ( d = 3 ). What is the 10th term?
Using the formula for the ( n )-th term:
[ a_{10} = a + (10 - 1)d ] [ a_{10} = 5 + (10 - 1) \cdot 3 ] [ a_{10} = 5 + 9 \cdot 3 ] [ a_{10} = 5 + 27 ] [ a_{10} = 32 ]
So, the 10th term of the AP is 32.
Example 2: Sum of the First ( n ) Terms
Using the same AP from Example 1, let's find the sum of the first 10 terms.
[ S_{10} = \frac{10}{2} [2 \cdot 5 + (10 - 1) \cdot 3] ] [ S_{10} = 5 [10 + 27] ] [ S_{10} = 5 \cdot 37 ] [ S_{10} = 185 ]
Therefore, the sum of the first 10 terms of the AP is 185.
Example 3: Finding the Common Difference
Suppose we have an AP with the first term ( a = 7 ) and the 5th term ( a_5 = 19 ). What is the common difference?
Using the formula for the ( n )-th term:
[ a_5 = a + (5 - 1)d ] [ 19 = 7 + 4d ] [ 4d = 19 - 7 ] [ 4d = 12 ] [ d = \frac{12}{4} ] [ d = 3 ]
The common difference ( d ) is 3.
Example 4: Finding the Number of Terms
If the first term of an AP is 2 and the sum of the first ( n ) terms is 110, and the common difference is 2, how many terms are there?
Using the sum formula:
[ S_n = \frac{n}{2} [2a + (n - 1)d] ] [ 110 = \frac{n}{2} [2 \cdot 2 + (n - 1) \cdot 2] ] [ 110 = \frac{n}{2} [4 + 2n - 2] ] [ 110 = \frac{n}{2} [2 + 2n] ] [ 110 = n(1 + n) ]
This is a quadratic equation that we can solve for ( n ):
[ n^2 + n - 110 = 0 ]
Solving this quadratic equation, we find that ( n = 10 ) (since ( n ) must be positive).
Therefore, there are 10 terms in the AP.
Arithmetic progressions are fundamental in various mathematical contexts and real-world applications, such as calculating installments, constructing staircases, and many other scenarios where a regular interval or spacing is involved. Understanding the concept of AP and its formulas is crucial for solving problems in exams and practical situations.