Question

For what value of p,
\[2p{\text{ }} - {\text{ }}1,{\text{ }}7\] and \[3p\] are three consecutive terms of an AP?

Answer 1

Hint: To solve the question, i.e., to find the values of p, we will first consider that \[2p{\text{ }} - {\text{ }}1,{\text{ }}7\] and \[3p\] are three consecutive terms of an AP. Then on applying one of the conditions of AP, i.e., half of the sum of the first term and the last term is equal to the value of middle term, after that on putting the values in the condition, we will make a linear equation, and hence after solving that equation we will get our required answer.

Complete step-by-step answer:
We need to find the value of p, so that \[2p{\text{ }} - {\text{ }}1,{\text{ }}7\] and \[3p\] will be three consecutive terms of an AP.
So, we will consider that \[2p{\text{ }} - {\text{ }}1,{\text{ }}7\] and \[3p\] are three consecutive terms of an AP.
We know that, in an AP, the half of the sum of the first and last term is equal to the value of the middle term.
So, here we have, first term $ = 2p - 1$ , last term $ = 3p$ and middle term $ = 7$
On putting the values in the above-mentioned condition, we get
\[\dfrac{{2p - 1 + 3p}}{2} = 7\;\]
\[
\Rightarrow 5p - 1 = 14 \\
\Rightarrow 5p = 14 + 1 \\
\Rightarrow 5p = 15 \\
\Rightarrow p = \dfrac{{15}}{5} \\
\Rightarrow p = 3 \\
\]
Thus, the value of p is \[3.\]

Note:In the solutions above, we have considered that \[2p{\text{ }} - {\text{ }}1,{\text{ }}7\] and \[3p\] are three consecutive terms of an AP. We can check whether the given terms are in AP or not.
So, on putting the value of p, we get the following terms,
\[
2p - 1{\text{ }} = {\text{ }}2\left( 3 \right) - 1{\text{ }} = {\text{ }}5 \\
7 \\
3p{\text{ }} = {\text{ }}3\left( 3 \right){\text{ }} = {\text{ }}9 \\
\]
So, the terms are \[5,7,9.\]
Now, we know that, if the terms are in AP, then the common difference between the two consecutive terms is the same through-out the AP.
So, \[7 - 5{\text{ }} = {\text{ }}2\]
\[9 - 7{\text{ }} = {\text{ }}2\]
Thus, \[5,7,9\] are in AP.