Proof of Finite Arithmetic Series Formula

As we know that,

S_n = n/2(a₁ + a_n)

As we know at S_n  is the sum of the arithmetic series of ‘n’ terms, therefore, we can display it like below: 

S_n = 1 + 2 + 3….+ (n-1) + n

S_n = n + (n-1) +…+ 3 + 2 + 1 

Adding both the Sn series we get the following result: 

 2S_n = n+1 + 2+(n-1) + 3+(n-2)…..

2S_n = n(n+1)

S_n = n(n+1)/2

In the above formula, we find that sum of every term on the left-side turn out to be ‘n+1’ for ‘n’ times. 
Hence, we have proved,

S_n = n(n+1)/2 

An ordered list of numbers is called a To findsequence. Each number in the sequence is called a term. The sequence usually has patterns that allow us to predict what the next term of the sequence will be. An arithmetic series is the sum of sequences in which each term is computed from the previous one by adding and subtracting a constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of consecutive terms, the second number is found by adding a constant number to the previous one. ,

In Arithmetic Series/Progression we come across three terms which are: 

• Common difference(d)
• n^th term(a_n)
• Sum of the first n terms(S_n)

Here, all the above three terms represent the property of the arithmetic progression. 

To find the common difference in the arithmetic progression the following procedure is followed: 

 d = a₂ – a₁ = a₃ – a₂ = a₄ – a₃….. = a_n-a_n-1

where,

• a₁, a₂, a₃….a_n  are the terms of the series
• “d” is the common difference which can be positive, negative or zero

  Also, an arithmetic progression can also be written in the form of common difference as shown below: 

a, a + d, a + 2d, a + 3d, a + 4d, ………. , a + (n – 1)d

Where, “a” is the first term of series. 

There are two major formulas while reading arithmetic progression: 

• nth term of Arithmetic Series 
• Sum of first n Terms 

nth term of Arithmetic Series

The formula for nth term is,

a_n = a + (n−1)d

where,

• a is the first term
• d is the common difference
• n is the number of terms
• a_n is the nth term