Question

The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then find the common difference of the progression.
A. $2$
B. $3$
C. $\dfrac{3}{2}$
D. $-1$

Answer 1

Hint: In this question, we are to find the common difference in the given arithmetic series. Here we have provided with some of the terms and their relation with one another. By comparing those terms with the general terms of the arithmetic series, we get the required values.

Formula Used: An arithmetic series is written as
$a+(a+d)+(a+2d)+...+(a+(n-1)d)+...$
Here the first term is represented by $a$ and the common difference is represented by $d$.
The $nth$ term of an arithmetic series is written as
${{t}_{n}}=a+(n-1)d$
The sum of the arithmetic series of $n$ terms is
${{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right]$
The common difference is calculated by the formula $d={{a}_{n}}-{{a}_{n-1}}$

Complete step by step solution: Consider the terms of the series with arithmetic progression as
$a+(a+d)+(a+2d)+...+(a+(n-1)d)+...$
The given terms are:
The first term: $a$
The third term: $(a+2d)$
The fourth term: $(a+3d)$
The seventh term: $(a+6d)$
It is given that,
The fourth term of the series is three times the first term i.e.,
$\begin{align}
  & (a+3d)=3a \\
 & \Rightarrow 3d=3a-a=2a \\
 & \Rightarrow 3d=2a\text{ }...(1) \\
\end{align}$
The seventh term exceeds the twice of third term by one i.e.,
$\begin{align}
  & (a+6d)=2(a+2d)+1 \\
 & \Rightarrow a+6d=2a+4d+1 \\
 & \Rightarrow 6d-4d=2a-a+1 \\
 & \Rightarrow 2d=a+1\text{ }...(2) \\
\end{align}$
Then, on solving (2) and (1), we get
\[\begin{align}
  & 3d=2a\Rightarrow d=\dfrac{2}{3}a \\
 & \Rightarrow 2\left( \dfrac{2}{3}a \right)=a+1 \\
 & \Rightarrow \dfrac{4}{3}a-a=1 \\
 & \Rightarrow \dfrac{a}{3}=1 \\
 & \therefore a=3 \\
\end{align}\]
On substituting the obtained value in (1), we get
$\begin{align}
  & 3d=2(3) \\
 & \Rightarrow 3d=6 \\
 & \therefore d=2 \\
\end{align}$

Option ‘A’ is correct

Note: In such type of question, we have to consider the given terms with their general form and by simplifying them, we get the required values.