CBSE Class 10 Mathematics Arithmetic Progressions Notes Set 04

Sequence and Arithmetic Progression

A sequence is a list of numbers arranged in a definite order by some rule. Each number is called a term. A sequence is finite if it has a last term and infinite if it goes on without end.

An arithmetic progression (AP) is a sequence in which each term, after the first, is obtained by adding the same fixed number to the term before it. This fixed number is the common difference d, and it can be positive, negative or zero. If the first term is a, an AP looks like a, a + d, a + 2d, a + 3d, and so on.

The common difference is found as d = a_k+1 − a_k for any k. Examples are 147, 148, 149 (d = 1), 950, 900, 850 (d = −50) and 5, 5, 5 (d = 0).

The nth term

The nth term of an AP with first term a and common difference d is a_n = a + (n − 1)d.

The nth term from the end of a finite AP with last term l is l − (n − 1)d. A useful link is a_n = S_n − S_n−1.

Sum of the first n terms

1. S_n = \( \dfrac{n}{2} \)[2a + (n − 1)d]

2. S_n = \( \dfrac{n}{2} \)(a + l), where l is the last term. Use this form when the last term is given.

3. The sum of the first n natural numbers is \( \dfrac{n(n+1)}{2} \).

Handy properties

1. Three numbers a, b, c are in AP if and only if 2b = a + c.

2. To take three terms of an AP whose sum or product is given, use a − d, a, a + d.

3. To take four terms, use a − 3d, a − d, a + d, a + 3d.

4. To check if a number is a term of an AP, find n from the formula. It is a term only if n is a positive whole number.

Worked examples

Example 1. Find the 15th term of the AP 3, 7, 11, 15, ... Here a = 3 and d = 4, so a₁₅ = 3 + 14 × 4 = 59.

Example 2. Find the sum of the first 20 terms of the AP 5, 8, 11, ... Here a = 5 and d = 3, so S₂₀ = \( \dfrac{20}{2} \)[10 + 19 × 3] = 10 × 67 = 670.

Example 3. How many multiples of 4 lie between 10 and 250, and what is their sum? The multiples are 12, 16, ..., 248. From 248 = 12 + (n − 1)4 we get n = 60. Their sum is \( \dfrac{60}{2} \)(12 + 248) = 30 × 260 = 7800.

Always write the values of a, d and n first, then substitute in the formula. This keeps the working neat and earns method marks.

Quick Check Questions