Arithmetic Series - Sum of n terms

Arithmetic Series

A sequence is the set of the outputs of a function defined from the set of natural numbers to the set of real numbers or complex numbers. If the codomain of the function is the set of real numbers, it is called a real sequence and if it is the set of complex numbers on the other hand, it is called a complex sequence. For example: 1, 4, 9, 16, 25, 36, 49 ……….625 represents a sequence of squares of natural numbers till 25.

In very simple terms, a sequence is an ordered set of numbers. A sequence is denoted using braces. The sequencearithmetic seriesrepresents all the natural numbers. A sequence can be finite or infinite depending upon the number of terms it can have.

Now that we know what a sequence is let us learn about series. Sequence and series are very often confused with each other. Series are derived from sequences.

A series is defined as the sum of the terms of a sequence. It is denoted by

arithmetic series 1

Where ai is the ith term of the sequence and I is a variable. ∑ is a symbol which stands for ‘summation’. It was invented by Leonard Euler, a Swiss mathematician.

The meaning of the above expression written using summation is:

arithmetic series 2

Let us now discuss some special arithmetic series and their sum.

  • 1 + 2 + 3 + 4 + ………. + n

This arithmetic series represents the sum of n natural numbers. Let us try to calculate the sum of this arithmetic series.

The difference between the sum of n natural numbers and sum of (n – 1) natural numbers is n, i.e.

Sn – Sn-1 = n

Let us now proceed by taking the difference of sum of n natural numbers and sum of (n -2) natural numbers and so on,

Sn – Sn-2 = n + (n – 1) = 2n -1

Sn – Sn-3 = n + (n – 1) + (n – 2) = 3n – (1 + 2)

Sn – Sn-4 = n + (n – 1) + (n – 2) + (n – 3) = 4n – (1 + 2 + 3)

Proceeding in the same manner, the general term can be expressed as:arithmetic series 4

According to the above equation the nth term is clearly kn and the remaining terms are sum of natural numbers preceding it.

Now if k = n, equation 1 can be written as:arithmetic series 5

Since S0 is 0 the above sum becomesarithmetic series 6

Also  sigma (p to n-1) {p} is one n short of Sn

Therefore,arithmetic series 7

Substituting the value of equation (3) in (2), we getarithmetic series 8

  • 12 + 22 + 32 + 42 + ………. + n2

This arithmetic series represents the sum of squares of n natural numbers. Let us try to calculate the sum of this arithmetic series.

To prove this let us consider the identity p3 – (p – 1)3 = 3p2 – 3p + 1. In this identity let us put p = 1, 2, 3…. successively, we get

23 – (1 – 1)3 = 3(1)2 – 3(1) + 1

23 – (2 – 1)3 = 3(2)2 – 3(2) + 1

33 – (3 – 1)3 = 3(3)2 – 3(3) + 1

………………………………………..

………………………………………..

………………………………………..

n3 – (n – 1)3 = 3(n)2 – 3(n) + 1

Adding both the sides of the equation, we getmis

We have already calculated the sum of n natural numbers asarithmetic series 10

Substituting this value n equation (4), we getarithmetic series 11

This represents the sum of squares of natural numbers using the summation notation. It can be simplified as:

arithmetic series 12

  • 13 + 23 + 33 + 43 + ………. + n3

This arithmetic series represents the sum of cubes of n natural numbers. Let us try to calculate the sum of this arithmetic series.

To prove this let us consider the identity (p + 1)4 – p4 =4p3 + 6p2 + 4p + 1. In this identity let us put p = 1, 2, 3…. successively, we get

24 – 14 =4(1)3 + 6(1)2 + 4(1) + 1

34 – 24 =4(2)3 + 6(2)2 + 4(2) + 1

44 – 34 =4(3)3 + 6(3)2 + 4(3) + 1

……..………………………………………..

………..……………………………………..

…………..…………………………………..

(n + 1)4 – n4 =4n3 + 6n2 + 4n + 1

Adding both the sides of the equation, we getmis 2

We have already calculated the sum of n natural numbers and sum of squares of n natural numbers asarithmetic series 14

Substituting these values in equation (4) and simplifying, we getarithmetic series 15

This represents the sum of squares of natural numbers using the summation notation. It can be simplified as:arithmetic series 16

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Practise This Question

The sum of the series  12.2 +  22.3 +  32.4 + ........ to n terms is