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Arithmetic Sequence Formula: How to Find General Term

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Arithmetic Sequence (Progression): Definition and Examples

We saw in the definition of an arithmetic sequence that we can write the following term in terms of the current term according to the following formula:

so that: is the next term;

and is the current term;

and is the common difference.

So, as you notice through this formula that one of the two terms is written in terms of the other. But, can we write the term in terms of only?

This is what we will answer in this lesson.

Let's take the previous formula and do some calculations:

This formula that we get is called the general term of an arithmetic sequence. Through it we can calculate the value of any term in position directly.

Note:

Let an arithmetic sequence with common difference . The term general of is:

Let's take some examples

Example1:

We consider the arithmetic sequence:

Find 20th term

Solution:

First, we must find

Thus,

Example2:

Let an arithmetic sequence with and . Find

Solution:

First, we must find

Thus,

Example3:

Et an arithmetic sequence with and .

Is 59 a term belongs to the arithmetic sequence ?

Solution:

In order 59 to be a term to belong to the arithmetic sequence , it must satisfy the general term formula such that is a natural integer.

We have:

Since 15 is a natural number, then, 59 is the 15th term in the sequence .

Example4:

Find the number of terms in the arithmetic sequence:

Solution:

Here we have a finite arithmetic sequence, where the common difference d is 3, and the first item is . We use the general term formula to calculate the number of terms in this sequence.

So this sequence contains 31 terms.

The general term formula enables us to calculate the value of nth term, if we reformulate it further, we get another formula that calculates the number of terms in a finite arithmetic sequence.

Note:

To determine the number of terms for a finite arithmetic sequence, we use the following formula:

Whereas: is the first term, is the last term and is the common difference.

Example5:

How many two-digit numbers are divisible by 3?

Solution:

The sequence of two-digit numbers divisible by 3 is a finite sequence. The first term of this sequence is 12 (the smallest two-digit number divisible by 3), and the last term of this sequence is 99 (the largest two-digit number divisible by 3).

The sequence is:

It's clear that is an arithmetic sequence with a common difference .

So that:

Therefore, athe sequence consisting of 30 terms of two numbers are divisible by 3.