Chapter 10: Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms remains constant. In this chapter, we master finding the $n^{\text{th}}$ term, sum of $n$ terms, and word problems based on A.P.

1. What is an A.P.?

An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant is called the Common Difference ($d$).

Standard Form

If the first term is $a$ and the common difference is $d$, the AP is written as:

$$a, \quad a+d, \quad a+2d, \quad a+3d, \quad \dots$$

2. The $n^{\text{th}}$ Term of an A.P.

$$t_n = a + (n - 1)d$$

Practice Problems — $n^{\text{th}}$ Term

Q1. Find the 20th term of the A.P. $9, 13, 17, 21, \ldots$

Solution: $a = 9, d = 4 \implies t_{20} = 9 + (19)(4) = 85$. Ans: 85

Q2. Which term of the A.P. $3, 8, 13, 18, \ldots$ is $78$?

Solution: $3 + (n-1)5 = 78 \implies 5(n-1) = 75 \implies n = 16$. Ans: 16th term

3. Sum of First $n$ Terms of an A.P. ($S_n$)

$$S_n = \frac{n}{2} \left[ 2a + (n - 1)d \right] \quad \text{OR} \quad S_n = \frac{n}{2} (a + l)$$