What is an Arithmetic Progression (AP)?
Imagine a sequence of numbers where the difference between any two consecutive numbers is always the same. That's an arithmetic progression! This constant difference is called the common difference (d).
Key Terms
- a: The first term in the sequence.
- d: The common difference between consecutive terms.
- n: The number of terms in the sequence.
- a<sub>n</sub>: The nth term in the sequence.
- S<sub>n</sub>: The sum of the first n terms.
General Form of an AP
An arithmetic progression takes the following form:
a, a + d, a + 2d, a + 3d, ...
Important Formulas
- nth term (a<sub>n</sub>): a<sub>n</sub> = a + (n - 1) * d
- Sum of n terms (S<sub>n</sub>): S<sub>n</sub> = (n/2) * [2a + (n - 1) * d] OR S<sub>n</sub> = (n/2) * [a + a<sub>n</sub>] (when the last term is known)
Let's Solve Some Examples!
Example 1: Finding the nth term
- Problem: Find the 10th term of the arithmetic progression: 3, 7, 11, 15...
- Solution:
- Here, a = 3 (first term) and d = 4 (common difference: 7 - 3 = 4)
- Using the formula: a<sub>n</sub> = a + (n - 1) * d
- a<sub>10</sub> = 3 + (10 - 1) * 4 = 3 + 9 * 4 = 3 + 36 = 39
- Therefore, the 10th term is 39.
Example 2: Finding the sum of n terms
- Problem: Find the sum of the first 20 terms of the arithmetic progression: 2, 5, 8, 11...
- Solution:
- Here, a = 2, d = 3 (5 - 2 = 3), and n = 20
- Using the formula: S<sub>n</sub> = (n/2) * [2a + (n - 1) * d]
- S<sub>20</sub> = (20/2) * [2 * 2 + (20 - 1) * 3] = 10 * [4 + 19 * 3] = 10 * 61 = 610
- Therefore, the sum of the first 20 terms is 610.
Example 3: Finding the number of terms
- Problem: How many terms are there in the AP 7, 13, 19, ...., 205?
- Solution:
- Here, a = 7, d = 6 (13 - 7 = 6), and a<sub>n</sub> = 205
- Using the formula a<sub>n</sub> = a + (n - 1) * d
- 205 = 7 + (n - 1) * 6
- 198 = (n - 1) * 6
- n - 1 = 33
- n = 34
- Therefore, there are 34 terms in this AP.
Let me know if you'd like more examples or have any questions!
