Arithmetic Progression: Overview, Questions, Preparation

A progression is a special sequence type for which the nth term formula can be obtained. With easy and simple formulae, arithmetic progression has been the most famous sequence in mathematics. Any sequence or series in the order wherein the difference between any consecutive number is constant is referred to as arithmetic progression. Moreover, in the case of any set of consecutive odd and even, the difference would be two. 

General Equation

For a sequence, a_n where the common difference is given by d and the first term is a₁. The equation would be:

a_n=a₁+(n-1)d

Key properties of Arithmetic Progression

Following are some of the key properties of an arithmetic progression:

1. If a,b,c in any random series are in A.P, then 2b = a + c.

2. The common difference would be ‘a’ for a sequence in A.P, where the n^th in the sequence is of the form an + b.

3. A sequence can also be in A.P, if a non-zero constant number divided, multiplies, adds to, or gets subtracted from each term of the series in A.P.

4. An arithmetic progression is a decreasing sequence if the common difference is negative, i.e. dn-1> a_n

5. An arithmetic progression is an increasing sequence if the common difference is positive, i.e. d>0, and satisfies the condition a_n-1n.

Sums of Arithmetic Progressions

In an AP with common difference ‘d’ and the first term ‘a’, the sum of the first n terms is given by:

S = n/2[2a + (n − 1) × d]

The syllabus of class 10 maths presents arithmetic progression in chapter 5 under unit II Algebra. The chapter holds 5 marks in the board examinations and covers topics such as the derivation of the nth term and sum of the first n terms of an A.P.

The syllabus of class 11 maths presents arithmetic progression in the chapter 7 sequence and series from unit Algebra. The unit holds crucial 30 marks and covers G.P, arithmetic means, and other pivotal topics.

1. Subba Rao started work in 1995 at an annual salary of Rs 5000 and received an increment of Rs 200 each year. In which year did his income reach Rs 7000?

Solution:

Given, the salary increases by 200 every year

Therefore the series would be,

5000,5200,5400……..,an

a=5000, d=200, an= 7000

Thus,

a_n = a+(n−1) d

7000 = 5000 + (n-1) 200

n = 11

The salary would be 7000 in the 11th year.

2. Find the sum of the first 22 terms of an AP in which d = 7 and the 22^nd term is 149.

Solution:

Given, d = 7, a ₂₂= 149

From the formula, a = 2

Now, 

S_n= n/2(a+an)

S_n= 22/2 (2 + 149)

S₂₂ = 1661

3. Write the first four terms of the A.P. when the first term a and the common difference where a = 10, d = 10.

Solution:

Given, a = 10, d = 10

Therefore, the series would be 10,20,30,40,50,...

First four terms are 10,20,30,40,

Q: What is the need for Arithmetic Progression?

Q: Give some instances of A.P in our daily life?

Q: What are the other types of progressions in Maths?

Q: What is the formula for recursive AP?

Q: What is the formula for Explicit AP?