Arithmetic Progression Important Tips

When a successive order of a number is arranged in a number series, that successive number is in A.P. Still, the difference between each term of the number series and the preceding number in the number series should be constant, tn-1 – tn = constant. This constant is known as the common difference (c.d) in the number series, and the common difference is denoted by “d” in A.P. 

Arithmetic Progression is abbreviated as “A.P”. In addition, remember that common differences can be positive, negative or zero and depend upon the series. 

For, e.g.:-

A series of natural numbers was given to prove that the common difference of each term of the series and the preceding number in A.P is constant:-

1,2,3,4,5,6,7

2-1= 1

3-2= 1

4-3= 1

5-4= 1

6-5= 1

7-6= 1

So, here you get a common difference of 1 in all. As it shows, the series is constant. 

Essential Terms of Arithmetic Progression

There is an essential tip for the student before starting this topic that they must know some important terms which are used in A.P. They are:

• The common difference in the A.P (A.P) series is denoted by “d”.

• The first term of the A.P (A.P) series is denoted by “a”.

• The nth term of the series is denoted by “an“.

• The sum of the first n terms in A.P (A.P) is denoted by ” Sn“.

Some Important Formulas of Arithmetic Progression:

The Arithmetic Progression topic formula plays an important role in solving the question of series. However, Let us take a series of A.P:-

a1, a2, a3, a4, a5, a6, a7,……………..an.

• The general formula of the A.P is a, a + d, a + 2d, a + 3d, …..

• In A.P the formula of the nth term is an 

 an =a+(n – 1)×d.

• The Sum of n terms in the A.P series is 

S = n/2 [2a+(n−1)×d].

• In A.P, the sum of all the terms in a finite A.P and the last term ” l ” is n/2(a + l). 

• To find the nth term of the series, the formula is Tn=Sn – Sn-1. However, Tn is the nth term of the series. 

(However, a is the First term in series, d is the common difference in series, n is the number of terms in series, and an is the nth term in series). 

Also, the middle term of an Arithmetic progression importance is given below.

Some Important Tips to solve  Arithmetic Progression

In series, the middle term of an Arithmetic progression importance is there are three numbers in a series. So, the middle one in the series is known as Arithmetic Mean. However, if the series is a, b, and c. Where b is the middle term of the series or Arithmetic Mean. So, the formula will be:-

                              b=(a+c)/2 .

If we consider the mth term of A.P series is n and the nth term was considered m in series. So, we considered the common difference between the terms would be -1, and the (m+n)th of AP  will be zero.

Illustration:

 If there is an A.P series given, the 12th term of A.P is 6, and the 6th term is 12. So, what will be the 18th term of an A.P? 

Given, 

The 12th term of the series is 6

6th term of the series is 12

So, a12  = 6

      a6 = 12

However, 

a+11d=6…… (i) 

a+ 5d=12…….(ii) 

(i) – (ii), we get

6d = -6

d = -1

Now put the value of d in equation (i), now we get, 

a= 11+6

So, the value of a=17, 

a18 = a+17d

a18= 17+17(-1) = 0,

Here, the value of n is -1, so the value of (m+n)th term is zero. 

Conclusion

Arithmetic Progression is a brief topic in which you have to solve the questions carefully as all the questions contain different conditions. However, there is one more critical tip before solving the A.P topic; we have to carefully learn all the formulas and their abbreviation because these are the basics of the topic and the whole case depends on these formulas only. 

As this topic also helps to solve the questions of trigonometry, algebra, number system, and also there are many chapters available in which there is a use of the A.P formula, through which we can solve those questions quickly without making any mistakes.