Arithmetic Sequence Formula

Arithmetic Sequence Formula

In mathematics, a sequence has several applications. It denotes an enumerated collection of things in which repeats are permitted in some defined manner. It is composed of members or phrases, much like a set. The number of items in a finite sequence is known as its length or the number of terms. Many times, we may generate a series from sequences. There are several popular sequences. One such series is the Arithmetic series. This topic will teach the learner about it as well as the Arithmetic Sequence formula, complete with examples. Let’s learn about a fascinating topic!

What Is the Arithmetic Sequence Formula?

Arithmetic sequences are defined in two ways. It may be defined as a “sequence where the differences between every two successive terms are the same” (or) a sequence in which “every term is obtained by adding a fixed number (positive or negative or zero) to its previous term” . The following is an arithmetic series, with each phrase created by adding a fixed number 4 to the preceding term.

Formula of Arithmetic Sequence

The first term of an arithmetic series is a, and the common difference is d, where n is the number of terms. The AP takes the following general form: a, a+d, a+2d, a+3d, and so on, up to n words. We have many formulae linked with an arithmetic sequence for calculating the nth term, the sum of n terms of an AP, or the common difference of a particular arithmetic sequence.

The arithmetic sequence formula is given as,

  • Nth Term: an = a + (n-1)d
  • Sn = (n/2) [2a + (n – 1)d]
  • d = an – an-1

Solved Examples Using Arithmetic Sequence Formula

Example 1: Using the arithmetic sequence formula, find the 23th term in the sequence 1, 5, 9, 13…

Solution:

To find: 23th term of the given sequence.

Since the difference between consecutive terms is the same, the given sequence forms an arithmetic sequence.

a = 1, d = 4

Using arithmetic sequence formula,

an = a1+ (n – 1) d

For 23th term, n = 23

an = 1 + (23 – 1)4

an = 1 + (22)4

an = 1 + 44

an = 45

Answer: 23th term in the sequence is 49.

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FAQs (Frequently Asked Questions)

Arithmetic series is nothing but the sum of a few or all terms of an arithmetic sequence. Thus, the arithmetic series formula is:

  • Sn = (n/2) [2a1 + (n – 1) d] [OR]
  • Sn = (n/2) [a1 + an]

Here, a1 is the first term of the arithmetic series and ‘d’ is its common difference.

To utilise the arithmetic sequence formula, start by identifying the sequence’s initial term (a1) and common difference. Substitute these into the appropriate formula (of the nth term or the total) and simplify.

The sum of the first n terms in an arithmetic sequence is given as, Sn = (n/2) [2a1 + (n – 1) d] where Sn = sum of n terms, a1 = first term, and d is the common difference.