Question

For what value of p are 2p +1, 13, 5p-3 three consecutive terms of AP ?

Answer 1

Hint:In an Arithmetic progression (AP) consecutive values differ by a same value. It can be used here by equating the difference in the second and the first term & third term and second term.

Complete step-by-step answer:
It is given that three consecutive terms of an AP are 2p +1, 13, 5p-3.
We are required to find the value of p if it is true.
First there is a need to understand what an Arithmetic progression (AP) is. We will start by defining it.
An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Difference here means the second minus the first. And similarly in other consecutive values.
We can simply start solving by making use of the definition of AP. i.e.
Second term – First term = Third term – Second term
\[13 - \left( {2p + 1} \right) = (5p - 3) - \left( {13} \right)\]
\[\begin{gathered}
  13 - 2p - 1 = 5p - 3 - 13 \\
   \Rightarrow 12 - 2p = 5p - 16 \\
   \Rightarrow - 7p = - 28 \\
   \Rightarrow p = 4 \\
\end{gathered} \]
Therefore, for P=4 the given terms are an arithmetic progression.

Note:This can be solved in a better and simpler method. In an AP if a, b and c are consecutive terms then by the principle of arithmetic mean \[2b = a + c\].
It can be applied in this question as terms are considered to be consecutive and in Arithmetic progression (AP).
Therefore, mathematically it can be shown as follows –
\[2 \times \left( {13} \right){\text{ }} = \left( {2p + 1 + 5p - 3} \right)\]
Now simplifying we get,
\[26{\text{ }} = \left( {7p - 2} \right)\]
\[28 = 7p \Rightarrow p = 4\]
Thus, this is a simpler and shorter method to solve the question. While applying this method we have to make sure that terms are consecutive and are of AP.