Question

What is the fifth term of the arithmetic sequence 2,….,8,……?

Answer 1

Hint: First we have to assume that the first term of the given sequence as ‘a’ and the common difference is ‘d’. Then we will find the common difference of the A.P. and then find the fifth term of the series using the basic formula which is given by $ {{a}_{n}}=a+\left( n-1 \right)d $
Where a = first term
d = common difference and n =number of terms

Complete step by step answer:
We have been given an arithmetic sequence 2,….,8,……
We have to find the fifth term of the given sequence.
Now, we know that the to find the fifth term of the given sequence we have a general formula $ {{a}_{n}}=a+\left( n-1 \right)d $
Here, a = first term of the sequence
So, we have $ a=2 $
Now, d = common difference which is the difference between two terms.
Now, we know that the second term is the sum of the first term and common difference and the third term is the sum of the second term and the common difference.
So, we have $ {{a}_{2}}={{a}_{1}}+d $ and $ {{a}_{3}}={{a}_{2}}+d $
We have given in the question $ {{a}_{3}}=8 $ and $ {{a}_{1}}=2 $
Substituting the values we get
 $ \Rightarrow {{a}_{2}}=2+d.........(i) $ and \[\Rightarrow 8={{a}_{2}}+d...........(ii)\]
Now, substituting the value from equation (i) we get
\[\begin{align}
  & \Rightarrow 8=2+d+d \\
 & \Rightarrow 8-2=2d \\
 & \Rightarrow 6=2d \\
 & \Rightarrow d=\dfrac{6}{2} \\
 & \Rightarrow d=3 \\
\end{align}\]
Now, the fifth term will be
 $ \begin{align}
  & \Rightarrow {{a}_{5}}=2+\left( 5-1 \right)3 \\
 & \Rightarrow {{a}_{5}}=2+\left( 4 \right)3 \\
 & \Rightarrow {{a}_{5}}=2+12 \\
 & \Rightarrow {{a}_{5}}=14 \\
\end{align} $

So, the fifth term of the given sequence is 14.

Note:
 If it is not given in the question that the given sequence is arithmetic then first we have to identify the series. Also, we know that the common difference in an arithmetic sequence is common between all terms. So, after finding the value of the common difference one can directly calculate the terms of the sequence by adding the value of the common difference to the previous term. For example- here in this question the common difference is 3.
So, the sequence will be
 $ \begin{align}
  & 2+3=5 \\
 & 5+3=8 \\
 & 8+3=11 \\
 & 11+3=14 \\
\end{align} $
The sequence will be 2,5,8,11,14 and the fifth term is 14.