Question

How do I find the common difference of the arithmetic sequence 5, 9, 13, 17,....?

Answer 1

Hint: This question belongs to the topic of sequence and series. In this question, we will first know how to find the common difference of any arithmetic sequence. After that, we will find the common difference of arithmetic sequence which is given in the question. After that, we will see another method to solve this question.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to find the difference of arithmetic sequence 5, 9, 13, 17,....
Let us know a formula for \[{{\text{n}}^{th}}\] term of arithmetic sequence.
The formula is: \[{{a}_{n}}=a+\left( n-1 \right)d\], where \[{{a}_{n}}\] is \[{{\text{n}}^{th}}\] term of the sequence, \[a\] is the first term, \[d\] is the common difference, and \[n\] is the total number of terms.
For finding common differences, we will use this formula.
In the given sequence, the last or fourth term is 17 and first term is 5 and total number of terms is 4, so using the above formula we can write
\[17=5+\left( 4-1 \right)d\]
\[\Rightarrow 17-5=\left( 3 \right)d\]
\[\Rightarrow 12=3d\]
\[\Rightarrow d=4\]
So, we got that the common difference of the given sequence is 4.

Note: As we can see that this question is from the topic of sequence and series, so we should have a better knowledge in that topic for solving this type of question easily. We should know the formulas of arithmetic sequence. The formulas are in the following:
\[{{a}_{n}}=a+\left( n-1 \right)d\]
\[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\]
Where,
\[{{a}_{n}}\] is the \[{{\text{n}}^{th}}\] term of arithmetic sequence, a is the first term of arithmetic sequence, d is the common difference of arithmetic sequence, n is total number of terms and \[{{S}_{n}}\] is sum of n terms.
We can solve this question by alternate method.
So, let us first find out the difference between the consecutive numbers.
Let us find the difference between 5 and 9.
9-5=4
Let us find the difference between 9 and 13.
13-9=4
Let us find the difference between 13 and 17.
17-13=4
So, we got that the difference between the consecutive numbers is the same. And, we know that the difference between the consecutive numbers is 4.
Hence, we have found the common difference of the arithmetic sequence 5, 9, 13, 17,....
The common difference is 4.