Question

How do you find the common difference of an arithmetic sequence?

Answer 1

Hint: To solve the question that is mentioned in the above question, there are mainly two different methods, first being: if you know the first and the last term of the sequence then just use the formula
\[\begin{align}
  & {{a}_{n}}={{a}_{0}}+(n-1)d \\
 & \Rightarrow 12={{a}_{0}}+\left( 5-1 \right)d \\
 & \Rightarrow 4=2d \\
 & \Rightarrow d=2 \\
\end{align} \\ \]
where \[{{a}_{n}}\] is the last term of the sequence, d is the common difference of the sequence, \[{{a}_{0}}\] being the first term of the sequence. Second method being if you know any two different terms of the sequence convert it into two equations with respect to \[{{a}_{0}}\]and then solve those simultaneous equations and find the value of d from those equations.

Complete step-by-step solution:
For the first method let us take an example of a sequence where first term being 1 and the last term being 10 where 10 is term is the last term
So by using the formula: \[{{a}_{n}}={{a}_{0}}+(n-1)d\]
We will substitute the values in the above formula and we will get it as
\[\begin{align}
  & \Rightarrow 10=1+\left( 10-1 \right)d \\
 & \Rightarrow 9=9d \\
 & \Rightarrow d=1 \\
\end{align}\]
So this way we can also solve another type of question when we are given a term in the sequence and also know what term is it in the sequence then again solve it in the way we solved the above question.
Now to solve another question in which we don’t know the first term but know two different terms in the question, for example: we know the 3rd term of the sequence to be 8 and the 5th term of the sequence to be 12, by using these values and substituting in the formula we will get:
\[\begin{align}
  & \Rightarrow 8={{a}_{0}}+\left( 3-1 \right)d \\
 & \Rightarrow 8={{a}_{0}}+2d......\left( 1 \right) \\
\end{align}\]
\[\begin{align}
  & \Rightarrow 12={{a}_{0}}+\left( 5-1 \right)d \\
 & \Rightarrow 12={{a}_{0}}+4d......\left( 2 \right) \\
\end{align}\]
Now we will subtract equation 1 from 2 and we will get:
\[\begin{align}
  & \Rightarrow 4=2d \\
 & \Rightarrow d=2 \\
\end{align}\]
And this is how we can calculate the common difference of an arithmetic sequence.

Note: There is a general mistake while doing this question i.e. sometimes we forget to check which term we are given, which will then not result in the correct common difference. So always remember to check which term has been mentioned in the question and what type is it i.e. is it the first or the second type that we just discussed in the question above.