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