Question

The general form of A.P. is
A. $ a,\,a + d,\,a - d,\,a + 2d.... $
B. $ a,\,a + d,\,a + 2d,\,a + 3d.... $
C. $ 2a,\,a + 2d,\,a - d,\,a + 2d.... $
D. $ a,\,a - d,\,a - d,\,a - 2d.... $

Answer 1

Hint: Here before solving this question we need to know the about AP: -
An AP is a progression (increasing or decreasing) whose common difference of consecutive numbers is the same.
Mathematically,
 $ {a_4} - {a_3} = {a_3} - {a_2} = {a_2} - {a_1} $
Where,
 $ \begin{gathered}
  {a_1} = {\text{first}}\,{\text{term}} \\
  {a_2} = {\text{second}}\,{\text{term}} \\
  {a_3} = {\text{third}}\,{\text{term}} \\
  {a_4} = {\text{forth}}\,{\text{term}} \\
\end{gathered} $

Complete step-by-step answer:
According to the question,
Consider the sequence $ a,\,a + d,\,a - d,\,a + 2d.... $
Where,
 $ \begin{gathered}
  {a_1} = a \\
  {a_2} = a + d \\
  {a_3} = a - d \\
  {a_4} = a + 2d \\
\end{gathered} $
Now
 $ \begin{gathered}
  {a_2} - {a_1} = (a + d) - a = d \\
  {a_3} - {a_2} = (a - d) - (a + d) = - 2d \\
  \because {a_3} - {a_2} \ne {a_2} - {a_1} \\
\end{gathered} $
Therefore, it is not an A.P.
Similarly, the sequence $ a,\,a + d,\,a + 2d,\,a + 3d.... $
Where,
 $ \begin{gathered}
  {a_1} = a \\
  {a_2} = a + d \\
  {a_3} = a + 2d \\
  {a_4} = a + 3d \\
\end{gathered} $
Now
 $ \begin{gathered}
  {a_2} - {a_1} = (a + d) - a = d \\
  {a_3} - {a_2} = (a + 2d) - (a + d) = d \\
  {a_4} - {a_3} = (a + 3d) - (a + 2d) = d \\
  \because {a_4} - {a_3} = {a_3} - {a_2} = {a_2} - {a_1} \\
\end{gathered} $
Therefore, it is an A.P
So, the correct answer is “Option B”.

Note: We must not get confused with AP and GP. Secondly while choosing the common difference it should be in the form of
 $ \begin{gathered}
  {a_1} = a \\
  {a_2} = a + d \\
  {a_3} = a + 2d \\
  {a_4} = a + 3d \\
\end{gathered} $
If we select
 $ \begin{gathered}
  {a_1} = a \\
  {a_2} = a + 2d \\
  {a_3} = a + 3d \\
\end{gathered} $
The resultant would not form an AP.