Question

Find the common difference in the series 4, 8, 12, 16, 20.
Also find the sum of terms.
(a) 8
(b) 3
(c) 4
(d) 2

Answer 1

Hint: We solve this problem by using the definition of Arithmetic progression. The general representation of A.P is given as
\[a,\left( a+d \right),\left( a+2d \right),............\]
Here, \['a'\] is the first term and \['d'\] is the common difference.
The common difference can be calculated by the difference between any two consecutive numbers in the series. Also the formula for sum of \['n'\] terms in A.P is given as
\[{{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\]
By using the above two formulas we find the common difference and sum of terms of given sequence

Complete step-by-step answer:
We are given that the sequence as
4, 8, 12, 16, 20
We know that the general representation of A.P is given as
\[a,\left( a+d \right),\left( a+2d \right),............\]
Here, \['a'\] is the first term and \['d'\] is the common difference.
We also know that the common difference is calculated by the difference between any two consecutive numbers in the sequence
By using the above condition we get the common difference of given sequence as
\[\begin{align}
  & \Rightarrow d=8-4 \\
 & \Rightarrow d=4 \\
\end{align}\]
Here, we can calculate the common difference by using the other terms as
\[\begin{align}
  & \Rightarrow d=16-12 \\
 & \Rightarrow d=4 \\
\end{align}\]
Now, we know that the sum of \['n'\] terms in A.P is given as
\[{{S}_{n}}=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\]
Now, we have the first term of sequence \[a=4\] and common difference \[d=4\] and the number of term \[n=5\]
By substituting these values in formula we get
\[\begin{align}
  & \Rightarrow {{S}_{5}}=\dfrac{5}{2}\left( 2\times 4+\left( 5-1 \right)4 \right) \\
 & \Rightarrow {{S}_{5}}=\dfrac{5}{2}\left( 24 \right) \\
 & \Rightarrow {{S}_{5}}=60 \\
\end{align}\]
Therefore the sum of terms of given sequence is 60 and the common difference of the sequence is 4
So, option (c) is the correct answer.

So, the correct answer is “Option (c)”.

Note: We can solve for the sum of terms in other ways also.
Here we have only 5 terms
So, we can add then directly to get sum of terms that is
\[\begin{align}
  & \Rightarrow S=4+8+12+16+20 \\
 & \Rightarrow S=60 \\
\end{align}\]
Therefore the sum of terms of given sequence is 60

We also have other formula for sum of terms of an A.P as
\[\Rightarrow {{S}_{n}}=\dfrac{n}{2}\left( {{a}_{1}}+{{a}_{n}} \right)\]
Here, \['{{a}_{1}}'\] is the first term and \['{{a}_{n}}'\] is the \[{{n}^{th}}\] term
By substituting the required values in above formula we get
\[\begin{align}
  & \Rightarrow {{S}_{5}}=\dfrac{5}{2}\left( 4+20 \right) \\
 & \Rightarrow {{S}_{5}}=\dfrac{5}{2}\times 24 \\
 & \Rightarrow {{S}_{5}}=60 \\
\end{align}\]
Therefore the sum of terms of given sequence is 60