Class 10 NCERT Maths Chapter 5: Arithmetic Progressions
1. Introduction
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by \(d\).
Example: \(2, 5, 8, 11, 14, \dots\) has \(d = 3\).
2. General Term of an AP
The \(n\)-th term (\(a_n\)) of an AP is given by: \[ a_n = a + (n - 1)d \] where:
- \(a\) = first term
- \(d\) = common difference
- \(n\) = term number
3. Sum of First \(n\) Terms
The sum of the first \(n\) terms (\(S_n\)) is: \[ S_n = \frac{n}{2}[2a + (n - 1)d] \] or \[ S_n = \frac{n}{2}(a + a_n) \]
4. Important Points
- If \(d > 0\), the AP is increasing.
- If \(d < 0\), the AP is decreasing.
- If \(d = 0\), all terms are equal.
5. Example 1
Find the 10th term of the AP: \(7, 10, 13, 16, \dots\)
Here: \(a = 7\), \(d = 3\), \(n = 10\)
\[
a_{10} = 7 + (10 - 1)\times 3 = 7 + 27 = 34
\]
6. Example 2
Find the sum of the first 20 terms of the AP: \(5, 8, 11, \dots\)
Here: \(a = 5\), \(d = 3\), \(n = 20\)
\[
S_{20} = \frac{20}{2}[2\times 5 + (20 - 1)\times 3]
\]
\[
S_{20} = 10[10 + 57] = 10 \times 67 = 670
\]