Chapter 5

Class 10 Mathematics - Chapter 5: Arithmetic Progressions
Chapter 5: Arithmetic Progressions
1. Introduction to Arithmetic Progression (AP)
Definition: An Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number $d$ to the preceding term except the first term.

General Form of an AP: $$a, a + d, a + 2d, a + 3d, \dots$$ where $a$ is the first term and $d$ is the common difference.

Common Difference ($d$): $$d = a_{k+1} - a_k$$ Note that $d$ can be positive, negative, or zero.
Example 1.1: Identifying First Term and Common Difference

For the AP: $3, 1, -1, -3, \dots$, write the first term $a$ and common difference $d$.

Solution:
First term $a = 3$
Common difference $d = a_2 - a_1$: $$d = 1 - 3 = -2$$ Thus, $a = 3$ and $d = -2$.

2. $n^{\text{th}}$ Term of an AP
Formula: The $n^{\text{th}}$ term $a_n$ (or general term) of an AP with first term $a$ and common difference $d$ is given by: $$a_n = a + (n - 1)d$$ If there are $m$ terms in the AP, $a_m$ represents the last term, often denoted by $l$.

$n^{\text{th}}$ Term from the End: $$a_n = l - (n - 1)d$$
Example 2.1: Finding a Specific Term

Find the $10^{\text{th}}$ term of the AP: $2, 7, 12, \dots$

Solution:
Here $a = 2$, $d = 7 - 2 = 5$, and $n = 10$.
Using the formula: $$a_n = a + (n - 1)d$$ $$a_{10} = 2 + (10 - 1)5$$ $$a_{10} = 2 + (9 \times 5)$$ $$a_{10} = 2 + 45$$ $$a_{10} = 47$$ The $10^{\text{th}}$ term is $47$.

Example 2.2: Finding the Number of Terms ($n$)

Which term of the AP: $21, 18, 15, \dots$ is $-81$?

Solution:
Here $a = 21$, $d = 18 - 21 = -3$, and $a_n = -81$.
Substitute values into $a_n = a + (n - 1)d$: $$-81 = 21 + (n - 1)(-3)$$ $$-81 - 21 = (n - 1)(-3)$$ $$-102 = (n - 1)(-3)$$ $$(n - 1) = \frac{-102}{-3}$$ $$n - 1 = 34$$ $$n = 35$$ Therefore, the $35^{\text{th}}$ term of the AP is $-81$.

3. Sum of First $n$ Terms of an AP
Formulas: The sum $S_n$ of the first $n$ terms of an AP is given by: $$S_n = \frac{n}{2} [2a + (n - 1)d]$$ If the last term $l$ of the finite AP is given, the sum can also be written as: $$S_n = \frac{n}{2} (a + l)$$
Example 3.1: Calculating the Sum of an AP

Find the sum of the first $22$ terms of the AP: $8, 3, -2, \dots$

Solution:
Here $a = 8$, $d = 3 - 8 = -5$, and $n = 22$.
Using the sum formula: $$S_n = \frac{n}{2} [2a + (n - 1)d]$$ $$S_{22} = \frac{22}{2} [2(8) + (22 - 1)(-5)]$$ $$S_{22} = 11 [16 + (21 \times -5)]$$ $$S_{22} = 11 [16 - 105]$$ $$S_{22} = 11 \times (-89)$$ $$S_{22} = -979$$ The sum of the first $22$ terms is $-979$.

Example 3.2: Finding $n$ when Sum ($S_n$) is Given

How many terms of the AP: $24, 21, 18, \dots$ must be taken so that their sum is $78$?

Solution:
Here $a = 24$, $d = 21 - 24 = -3$, and $S_n = 78$.
Substitute into the formula: $$S_n = \frac{n}{2} [2a + (n - 1)d]$$ $$78 = \frac{n}{2} [2(24) + (n - 1)(-3)]$$ $$156 = n [48 - 3n + 3]$$ $$156 = n [51 - 3n]$$ $$156 = 51n - 3n^2$$ Divide the equation by 3: $$n^2 - 17n + 52 = 0$$ Factorising by splitting the middle term: $$(n - 4)(n - 13) = 0$$ $$n = 4 \quad \text{or} \quad n = 13$$ Both values are acceptable because the sum of terms from $5^{\text{th}}$ to $13^{\text{th}}$ is zero. Therefore, $n$ can be $4$ or $13$.