Arithmetic Progression

An Arithmetic Progression is derived as a chain of positive or negative numbers. Each subsequent number is obtained by adding or subtracting a fixed number to the proceeding term in an AP.

Arithmetic Progression

Let us see the following series:

1. 41,43,45,47,49—-
2. 53,53,53,53,53—-
3. -4,-2,0,2,4—–
4. 1.1,1.2,1.3,1.4—-

All the list mentioned above is an example of A.P. because

1. Every following number is obtained by adding 2 to the preceding number.
2. Every following number is obtained by adding 0 to the preceding number.
3. Every following number is obtained by adding 2 to the preceding number.
4. Every following number is obtained by adding 0.1 to the preceding number.

Common Difference– The number which we add or subtract to obtain another number throughout any AP is called the common difference. It is represented by ‘d’. It can be positive or negative.

Term– Every number in the chain of an A.P. is called a term. It also can be positive, negative, fraction or decimal.

Types of A.P.

Two types of an A.P. are mentioned in mathematics:

1.Finite A.P.- If in an AP there are only a finite no. of terms, then such an A.P. is called finite A.P.

Example- 140,142,144,———160

2.Infinite A.P.- If in an AP there is not a finite no. of terms, then such A.P. is called infinite A.P.

Example- 111,122,133,144,……….

The general form of an A.P. 

If a is the first term of an A.P., and d is the common difference of an A.P., we can represent A.P. in this sequence:

a, a+d ,a+(d+d), a+(2d+d),………..

This is the simple way of representing an A.P., so if we have a first term (a) and difference (d) in any question, we can find an A.P.

Common differences in an A.P. 

If we find the difference between the second and first term of an A.P., we can find the common difference. In general,

A.P.= a1 ,a2, a3,—–, an

Then d = ak+1 – ak

Where ak+1 = ( k+1 )th term

And ak = k th term

Example – if we have a = 7 and d = 3 then A.P. must have

a1 = 7

a2 = 7+3 =10

a3 = 7+2×3 = 7+6= 13

a4 = 7+3×3 = 7+9 = 16 and so on…

nth term of an A.P.

The nth term of an AP = a+(n-1)d

Where a = first term, d = common difference, n = nth term

an = a+(n-1)d

By this formula, we can also find the last term of an A.P., which is represented by l

l = a+(n-1)d

If we have three terms, we can find the fourth term easily with these two formulas.

For example, if Sita has a salary of Rs 10000 with an annual increment of Rs 1000, what would be her salary after 15 years.

In this example, we do not need to find the A.P. because it will become a very long process and will be time-consuming. So to solve these types of questions and simplify, we have to find the formula of ‘nth’ term. It is as under: 

First-term = a

II term = a+d = a+(2-1)d

III term = a+2d = a+(3-1)d

And so on

Conclusion

An arithmetic progression contains a chain of numbers, with a common difference between all the numbers. A list of numbers a1, a2, a3,.. is an AP if the difference between all the terms gives the same value. The common form or generalisation of an AP is a, a+d, a+(d+d), a+(2d+d), and so on. In an AP, if first term a and common difference d, then the nth term is given by,

an = a+(n-1)d.