1. Introduction to Arithmetic Progression (AP)

A sequence of numbers is called an Arithmetic Progression (AP) if the difference between consecutive terms remains constant.

Example:

  • 2, 4, 6, 8, 10, ... (Common difference = 2)
  • 100, 95, 90, 85, ... (Common difference = -5)

The constant difference between consecutive terms is called the common difference (d).

2. General Form of an AP

An arithmetic progression can be written as:
a, a + d, a + 2d, a + 3d, ...

Where:

  • a = First term
  • d = Common difference
  • n = Number of terms

3. nth Term of an AP

The formula for the nth term (Tâ‚™ or aâ‚™) of an arithmetic progression is:


T_n = a + (n - 1) d

Example:

Find the 10th term of the AP: 3, 7, 11, 15, ...

  • a = 3, d = 7 - 3 = 4, n = 10

T_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39

Thus, the 10th term is 39.

4. Sum of First n Terms of an AP

The sum of the first n terms of an AP is given by:


S_n = \frac{n}{2} \times [2a + (n - 1) d]

Alternatively, if the last term (l) is given, then:


S_n = \frac{n}{2} \times (a + l)

Example:

Find the sum of the first 15 terms of the AP: 5, 10, 15, 20, ...

  • a = 5, d = 10 - 5 = 5, n = 15

S_{15} = \frac{15}{2} \times [2(5) + (15 - 1) \times 5]


= \frac{15}{2} \times [10 + 70] = \frac{15}{2} \times 80 = 600

Thus, the sum of the first 15 terms is 600.

5. Properties of Arithmetic Progression

  1. If each term of an AP is added/subtracted by a fixed number, the sequence remains an AP.
  2. If each term of an AP is multiplied/divided by a fixed number (non-zero), the sequence remains an AP.
  3. Any three consecutive terms of an AP can be taken as (a - d), a, (a + d).
  4. If an AP has an odd number of terms, then the middle term is the average of the first and last terms.

6. Applications of AP

  • Daily life problems like savings, salaries, and installment payments.
  • Finding missing terms in number patterns.
  • Predicting future values in sequences.

7. Solved Examples

Example 1: Find the number of terms

Find the number of terms in the AP: 7, 13, 19, ..., 205

  • a = 7, d = 13 - 7 = 6, l = 205
  • Using Tâ‚™ = a + (n - 1)d

205 = 7 + (n - 1) \times 6


198 = (n - 1) \times 6


n - 1 = 33 \Rightarrow n = 34

Thus, there are 34 terms in the AP.

Example 2: Find the middle term

Find the middle term of the AP: 10, 16, 22, ..., 82

  • a = 10, l = 82, d = 6
  • n is found using Tâ‚™ = a + (n - 1)d

82 = 10 + (n - 1) \times 6


72 = (n - 1) \times 6


n - 1 = 12 \Rightarrow n = 13

Since there are 13 terms, the middle term is T₇


T_7 = 10 + (7 - 1) \times 6 = 10 + 36 = 46

Thus, the middle term is 46.

8. Summary of Important Formulas

This covers Arithmetic Progression (AP) in detail with formulas, properties, and solved examples. Let me know if you need more practice questions!