Arithmetic Sequences: Definition, Formulas, and Solved Problems

Arithmetic Sequences

In mathematics, a sequence is a set of numbers that follows a specific order. Each number present in a sequence is called a term or member of this sequence. A sequence may be with finite or infinite terms. The finite sequence contains the last term, but an infinite sequence has not the last term.

The sequence has three main types: geometric, arithmetic, and Harmonic. However, we will confine ourselves to arithmetic sequences only. An arithmetic sequence or arithmetic progression (A.P) is a sequence where the common difference between of two consecutive terms is constant.

In this article, we will explain the arithmetic sequence with its examples. Then we will discuss the different formulas of an arithmetic sequence. After this, we will learn how to calculate the nth term and sum of the n term of arithmetic sequences with examples.

What is Arithmetic sequence (or Arithmetic progression)?

A sequence in which the common difference between every two – consecutive numbers are constant is called an arithmetic sequence. It is also known as arithmetic progression (A.P). Let’s learn it by example,

4, 7, 10, 13, 16, 19

Find common differences 7 – 4 = 3, 13 – 10 = 3, 16 – 13 = 3, 19 – 16 = 3. It is an arithmetic sequence because the common difference of every two consecutive numbers is constant that is 3. 

In other words, the next term can be obtained by adding or subtracting any fixed number in an arithmetic sequence. 2, 5, 7, 9, 11, 13… Is an arithmetic sequence because we added 2 to the previous term to obtain the next term.

Arithmetic sequence or Arithmetic progression (A. P) formulas to find the nth term

  1. Mathematically, an Arithmetic sequence can be written as

a, a + d,  a + 2d, a + 3d, … a + (n – 1)d  , n > 0

 Where “d” is a common difference and “n” is the number of the term. Like

  • First term of A.P = a1 = a + (1 -1) d = a ⇔ a1 – a= d
  • Second term of A.P = a2 = a + (2 -1) d = a + d ⇔ a2 – a1 = d
  • The third term of A.P = a3 = a + (3 -1) d = a + 2d ⇔ a3 – a2 = d

Similarly,

  • nth term of A.P  = an = a + (n -1) d ⇔ an – an – 1 = d
  • Common difference

d = an – an – 1

  • Nth term of A.P can also obtain by common difference formula

an – an – 1 = d ⇔ an = d + an – 1

  • Formula to find nth term of Arithmetic sequence arithmetic progression

an = a + (n – 1) d

  • Recursive formula to find the nth term of an arithmetic sequence or arithmetic progression

an = d + an – 1

Formula to find the sum of the n term of A.P

The sum of the n term (Sn) of the arithmetic sequence can be written as

Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + (a + (n – 1) d) ______ (i)

Re-write this sum in inverse as

Sn = an + (an – d) + (an – 2d) + (an – 3d) + … + (an – (n – 1) d) ______(ii)

Now add the equation (i) and (ii)

Sn = a + (a + d) + (a + 2d) + (a + 3d) + … + (a + (n – 1) d)

Sn = an + (an – d) + (an – 2d) + (an – 3d) + … + (an – (n – 1) d)

2 Sn = (a + an) + (a + an) + (a + an) + … + (a + an)

2 Sn = n (a + an)

Sn = (n / 2) {a + an} ____ (iii)

Also, we know that an = a + (n -1) d, by putting it in (iii),

Sn = (n / 2) {a + a + (n – 1) d}

Sn = (n / 2) {2a + (n – 1) d}

Sum of n term of A.P = Sn = (n / 2) {a + an} OR Sn = (n / 2) {2a + (n – 1) d}

Solved examples of the arithmetic sequence

Example 1:

Find the 7th term of A.P. if the first term is 5 and the common difference is 9.

Solution:

Here:

First term = a = 5

Common difference = d = 9

n = 7

As we know that, an = a + (n – 1) d

By putting the given value in formulas we have,

a7 = 5 + (7 – 1) 9

a7 = 5 + (6)9 = 5 + 54 = 59

An nth term calculator can be an alternate way to solve the problems of arithmetic sequence to find nth term.

Example 2:

Find the sum of the first 30 natural numbers.

Solution:

The first 30 natural numbers are 1, 2, 3, 4, 5… 30.

Here, the first term = a = 1, and the nth term of A.P is 30

The sum of n term = Sn = (n / 2) (a1 + an)

The sum of the first 30 natural numbers = S30 = (30 / 2) {1 + 30} = 15 (31) = 465

FAQs

How we can calculate the nth term of A.P?

By using a formula,

an = a + (n – 1) d

Where,

  • a = first term of A.P
  • n, n > 0= number of terms
  • d = common difference = an – an – 1

How to calculate the sum of n term in A.P?

The sum of n term = Sn = (n / 2) (a1 + an)

Here, a1 is the first term and an is the last term in A.P.

What is the sum of the first 50 natural numbers?

The sum of the first 50 natural numbers is equal to 1275.

Is any difference between arithmetic sequence and arithmetic progression?

No, there is no difference. Arithmetic progression (A.P) arithmetic sequences are the same.

Conclusion

In this article, we explained arithmetic sequence in detail with its examples. The formulas to calculate the nth term and sum of the n term of the arithmetic sequence are covered in this article. After this, we solved some examples of an arithmetic sequence.

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