Nth Term of Arithmetic Sequence

Let’s consider an example of Arithmetic Sequence 6, 16, 26, 36, 46, 56, 66, . . .. 

As we can see each term of this example can be represented in a similar form. Thus, the n^th term can be found easily by adding one less than n multiple of 10 to the first term of the sequence i.e., 6. 

Thus, the n^th term of the given example can be generalized as 6 + (n-1)×d.

In general, this is the standard explicit formula of an arithmetic sequence whose first term is, A, end, and the common difference is D is given as follows:

a_n = a + (n-1)×d

Read more: How to Find the Nth term of Arithmetic Sequence?

Recursive Formula for Arithmetic Sequence

The n^th term of an Arithmetic Sequence can be defined recursively as the next term can always be obtained by adding a common difference to the preceding term, the following derivation can be used to illustrate the same thing.

As we know, n^th term of the Arithmetic Sequence is given by 

a_n = a + (n-1)×d

thus, (n-1)^th term can be given by 

a_n-1 =  a + (n-1-1)×d

a_n-1 = a + (n-2)×d

Thus,  

a_n = a + (n-1)×d

⇒ a_n = a+(n-1-1+1)×d

⇒ a_n = a + (n-2+1)×d 

⇒ a_n = a + (n-2)×d + d

⇒ a_n = a_n-1 + d

Sum of terms in Arithmetic Sequence

Let’s sequence is given as a, a+d, a+2d, a+3d, ….. a+(n-1)d.

S_n = (n/2)(a + l) 

where,

• a is the first term
• l is the last term of the series and 
• n is the number of terms in the series

Replacing the last term l by the n^th term = a + (n – 1)d, we get

S_n = (n/2)(a + a + (n – 1)d)

OR

S_n = (n/2)(2a + (n – 1) x d)

Arithmetic Sequence is a type of sequence out of all sequences where each term of the sequence is related to the previous term of the sequence by a linear relation. A sequence is a collection of objects where all the terms follow an order or pattern by which the whole sequence can be identified. In the case of an Arithmetic Sequence, each term can be found by adding a constant to the preceding term of the Arithmetic Sequence, this constant sets the Arithmetic Sequence apart from the other sequences.

In this article, we will explore the concept of Arithmetic Sequence and various formulas associated with it. We will also learn about the various properties of Arithmetic Sequences.