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Table of contents
1.
Introduction 
2.
Arithmetic Progression
2.1.
Common Difference
2.2.
General Term
2.3.
Examples
3.
Sum of n terms in an A.P.
3.1.
Examples
4.
Properties of Arithmetic Progression
5.
FAQ
6.
Key Takeaways
Last Updated: Mar 27, 2024

Arithmetic Progression

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Introduction 

Progression is defined as a list of numbers that exhibit a specific pattern. It is also known as a sequence. Every (n+1)th term in a progression is obtained by applying a specific rule on its nth term. For example, 2, 4, 6, 8, ... is a progression because every number is obtained by adding 2 to its previous number. This pattern can vary from progression to progression.

In general, every progression term is defined by the nth term, denoted by an. For example, the nth term of a progression 2, 4, 6, 8, ... is an = 2n. Substituting n = 1, 2, 3, ... here, we will get the 1st, 2nd, and 3rd, .... terms.

In mathematics, there are mainly three types of progressions: 

  1. Arithmetic progressions
  2. Geometric progressions
  3. Harmonic progressions
     

This blog will discuss one of the types of progression,i.e., Arithmetic Progression.

Arithmetic Progression

An arithmetic progression is a sequence in which the difference between any two consecutive terms is constant. This constant difference in the arithmetic progression is known as the common progression difference and is generally denoted by ‘d’. 

 

 

Common Difference

The common difference is the difference between any two consecutive terms that can be easily calculated using the formula d = an+1 - an. In an A.P., this difference remains constant throughout the entire sequence.

Consider the following examples:

  1. Given a sequence: -1, 3, 7, 11, 15,...
    To calculate the common difference of the given sequence we follow the formula d= a- a1.i.e., 3 - (-1) = 4. Hence, the common difference in the given sequence is 4. Since this difference is the same among the entire sequence, the given sequence is an A.P. An increasing A.P. is a type of progression where the common difference is positive.
     
  2. Given a sequence: 25, 20, 15, 10, 5,...
    To calculate the common difference of the given sequence, we follow the formula d= a- a1.i.e., 20 - (25) = -5. Hence, the common difference in the given sequence is four. Since this difference is the same among the entire sequence, the given sequence is an A.P. A type of progression where the common difference is a negative number is called decreasing A.P.
     

General Term

An arithmetic progression is generally represented as a1, a2, a3,...., an. If the first term, generally denoted by ‘a’, and the common difference ‘d’ in any given arithmetic sequence is known, we can easily calculate the nth term using the given formula.

an = a + (n-1)*d

Here, an is known as the general term of the sequence. We can easily find the complete sequence if the general term for a sequence is known.

Derivation of the General Term

We have learned earlier that the common difference of an arithmetic sequence is d= an+1 - an

The given expression can also be written as an+1 = a+ d. Substitute n=1, 2, 3, …

The first term is represented by a and the common difference by d. The sequence will be a, a+d, a+ 2*d, ….. Following the pattern, the nth term will become a+ (n-1)*d.

Examples

  1. Given an A.P. whose fifth term is 34, the ninth term is 50. Find the first term of the sequence, the common difference, and the nth term.
    The fifth term of an A.P. is represented by a+ 4d = 34
    The ninth term of an A.P. is represented by a+ 8d = 50
    Solving the above two equations will give a = 18 and d= 4.
    The nth term for the A.P. will be a+ (n-1)*d. I.e., 18+ (n-1)*4.

     
  2. Given an A.P. whose first term is 2, the common difference is -2. Find the A.P.
    An A.P. is represented as a, a+d, a+2*d… Substituting the value of a and d in the above sequence will give us the desired A.P. The A.P. is 2, 0, -2, -4, …
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Sum of n terms in an A.P.

The sum of the first n terms of an arithmetic progression is given by

Sn= n/2 [2*a + (n-1)*d]

Here, Sn represents the sum of first n terms.

n is the number of terms in the sequence
a is the first term of the sequence.
d is the common difference of the sequence
 

Derivation of the sum of n terms

an = a, a+d, a+2*d, a+ 3*d, …
S= (a) + (a + d) + (a + 2*d) + (a + 3*d) + … + (a+ (n-1)*d)   — (1)

Sn = (a+ (n-1)*d) + … + (a + 3*d) + (a + 2*d) +  (a + d) + (a)  — (2)

Adding both equations (1) and (2) will give

2 Sn = (2*a + (n-1)*d) + (2*a + (n-1)*d) + (2*a + (n-1)*d) + … + (2*a + (n-1)*d)

Solving the above equation will give

2 Sn = n (2*a + (n-1)*d) that can also be written as

Sn = n/2 [2*a+(n-1)d].

An alternative formula to find the sum of the first n terms can also be written as 

Sn = n/2 [2*a+(n-1)d] => Sn = n/2 [(a) + (a + (n-1)d)].

Sn = n/2 [First Term + Last Term].

Examples

  1. Find the sum of an A.P. containing 12 terms whose first term is 3 and the last term is -15.
    The sum for the first n terms is given by Sn = n/2 [First Term + Last Term].
    Substituting the values in the given formula Sn = 12/2 [3 + (-15)].
    Sn = -72.
     
  2. Given an A.P. containing 4 terms whose sum is 24 and the last term is 12. Find the first term.
    Sn = n/2 [First Term + Last Term]
    24 = 4/2 [First Term + 12].
    Solving the above equation will give the First term = 0. The A.P. is 0, 4, 8, 12.

Properties of Arithmetic Progression

  • The sum of the kth term and the (n-k+1)th term is always constant.



 

  • Three numbers a, b, and c are said to be in an A.P. if they follow the equation 2*b = a+ c.
  • If each term of an Arithmetic Progression is added or subtracted from the same number, then the resulting sequence is also an A.P. with the same common difference.
  • If each term of an arithmetic progression is divided or multiplied by the same non-zero number, then the resulting sequence is also an arithmetic progression.
  • If the nth term of an A.P is a linear expression, then the sequence is an A.P.
  • If we select terms from an A.P in a regular interval, the selected terms will also form an A.P.
  • The graph of an A.P. is a straight line with the slope as the common difference.

FAQ

  1. What is the use of Arithmetic Progression?
    An arithmetic progression can be used to observe patterns in our day-to-day life.
     
  2. What are the types of Arithmetic Progression?
    An arithmetic progression is further divided into two parts: Finite A.P. and Infinite A.P.
     
  3. What is the relation between Arithmetic Mean, Geometric Mean, and Harmonic Mean?
    The relation between A.M., G.M., and H.M. is given by G2=A*H, where AM>= GM>= HM.
     
  4. Find the sum of the first 11 terms of an Arithmetic Progression whose third term is 4 and the seventh term is two more than thrice of its third term.
    Third term: a + 2d = 4  — (1)
    Seventh term: a + 6d = 3*4 + 2  — (2)
    Solving equations (1) and (2) will give a = -1 and d= 5/2 
    Sn= 12/2 [ 2(-1) + (10)*5/2] => Sn= 138.

Key Takeaways

In this article, we have extensively discussed Arithmetic Progression. An arithmetic progression is a sequence in which the difference between any two consecutive terms is constant. These sequences are used to find patterns in our day-to-day lives. 

Recommended Readings:

We hope that this blog has helped you enhance your knowledge regarding Arithmetic Progression, and if you would like to learn more, you can follow our guided path. Do upvote our blog to help other ninjas grow. 

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