Examples of arithmetic sequences. Here it’s all about the arithmetic sequence. Have a look!

### Arithmetic sequence

Well, it’s an old topic from high school. So I’ll not go into much detail.

Suppose I have a sequence like .

Here is the first term and is the common difference in the sequence.

Then this sequence is an arithmetic sequence.

There are also other sequences like geometric sequence, harmonic sequence and so on.

Now I’ll give some examples of arithmetic sequences.

#### Examples of arithmetic sequences

Here are some examples of arithmetic sequences.

Disclaimer: None of these examples is mine. I have chosen these from some book or books. The references are at the end of the post.

##### Example 1

According to Stroud and Booth (2013)* “The seventh and eleventh terms of an arithmetic sequence are and . Find the first term and the common difference.”

##### Solution

The standard form of an arithmetic sequence is

(1)

Here is the first term and is the common difference.

Also, I already know the seventh and eleventh terms of the sequence as and respectively.

So, first of all, I’ll put in equation (1). Thus I get

But in the example, the seventh term is .

(2)

Next, I’ll put in equation (1). Thus I get

But in the example, the eleventh term is .

(3)

Now I’ll solve equations (2) and (3) to get the values of and .

So, I’ll subtract equation (2) from equation (3).

Therefore I get

So it means .

Now I’ll substitute this value of in equation (2).

Therefore I get

Next, I’ll simplify it to get the value of .

Thus the value of is

So I can conclude that the first term and the common difference are and respectively.

This is the answer to this example.

Now I’ll go to my next example.

##### Example 2

According to Stroud and Booth (2013)* “If form three successive terms of an arithmetic sequence, find the next four terms.”

##### Solution

First of all, I’ll choose the general term of the arithmetic sequence, say .

Here is the first term and is the common difference. Let me choose three successive terms in this sequence.

So let them be and .

So that means,

But from the example, I already know that the three successive terms are .

(4)

(5)

(6)

Now my job is to find out the values of , and .

###### Step 1

First of all, I’ll add equations (4), (5) and (6).

Therefore I get

This gives

But from equation (5), I already knpw that

So this means

Thus it gives .

Therefore the three successive terms of the arithmetic sequence are . This gives the terms as

Thus I can say the first term is 3. the common difference is .

Hence I can say that the general form of this arithmetic sequence is

Now I already know the first three terms, that is .

Next, I’ll find out the values of

Thus it will be

So the next four terms in this sequence are 39, 51, 63 and 75.

This is the answer to this example.

Now I’ll go to the next example.

##### Example 3

According to Stroud and Booth (2013)* “If form three successive terms of an arithmetic sequence, show that also form three successive terms of another arithmetic sequence.”

##### Solution

Here are three successive terms of an arithmetic sequence. So I can say

(7)

Now I have to show that also form three successive terms of another arithmetic sequence. This means that I have to prove

So, first of all, I’ll simplify equation (7).

Thus it will be

So this means also form three successive terms of another arithmetic sequence.

Hence I have proved this statement.

This is the answer to this last example.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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