Question

For what value of \[a\] the terms \[2a-1,7,3a\] are three consecutive terms of an A.P?

Answer 1

Hint: We solve this problem by using the standard condition of three terms of an A.P We have the condition that if \[p,q,r\] are the three consecutive terms of an A.P then \[2q=p+r\] .By using the above condition to given terms we find the required value.

Complete answer:
We are given that the three consecutive terms of A.P as
\[2a-1,7,3a\]
We know that the condition that if \[p,q,r\] are the three consecutive terms of an A.P then
\[2q=p+r\]
By using the above condition to given terms of the A.P then we get
\[\begin{align}
  & \Rightarrow 2\left( 7 \right)=\left( 2a-1 \right)+\left( 3a \right) \\
 & \Rightarrow 14=5a-1 \\
\end{align}\]
Now, let us rearrange the terms in such a way that the variable terms come one side and constants on other side then we get
\[\begin{align}
  & \Rightarrow 5a=14+1 \\
 & \Rightarrow a=\dfrac{15}{5} \\
 & \Rightarrow a=3 \\
\end{align}\]
Therefore, we can conclude that the value of \[a\] for which the given terms are the 3 consecutive terms of A.P is 3 that is
\[\therefore a=3\]

Note:
We can solve this problem in another method.
We are given that the three terms of A.P as
\[2a-1,7,3a\]
We know that the common difference of A.P is given by the difference of two consecutive terms.
Let us assume that the common difference of A.P as \[d\]
Now, by using the definition of common difference for first two terms then we get
\[\begin{align}
  & \Rightarrow d=7-\left( 2a-1 \right) \\
 & \Rightarrow d=8-2a.....equation(i) \\
\end{align}\]
Now, let us use the common difference for second and third terms then we get
\[\Rightarrow d=3a-7.....equation(ii)\]
We know that the common difference will not change throughout the A.P
By using the above condition to equation (i) and equation (ii) we get
\[\begin{align}
  & \Rightarrow 8-2a=3a-7 \\
 & \Rightarrow 5a=15 \\
 & \Rightarrow a=3 \\
\end{align}\]
Therefore, we can conclude that the value of \[a\] for which the given terms are the 3 consecutive terms of A.P is 3 that is
\[\therefore a=3\]