Question

For what value of k will the consecutive terms $2k + 1,3k + 3$ and $5k - 1$ form an AP?

Answer 1

Hint: AP or Arithmetic Progression refers to a sequence of numbers in which the difference between two consecutive terms is the same all over the sequence. For example, if $2,5,8,11,.....$ this sequence is an AP, as the difference between the consecutive terms is equal to $3$ throughout the AP. Now, if three terms $a,b,c$ are in AP, then we can find the arithmetic mean of the three terms, by the formula,
$2b = a + c$
We will apply this formula to the terms given to us and find the required value of k in the given problem.

Complete step-by-step answer:
The three consecutive terms of the AP, given are,
$2k + 1,3k + 3,5k - 1$
So, let us name the terms as, $a = 2k + 1$
$b = 3k + 3$
$c = 5k - 1$
Now, applying the formula of arithmetic mean, that is, $2b = a + c$, and substituting the values of $a,b,c$ we get,
$2\left( {3k + 3} \right) = \left( {2k + 1} \right) + \left( {5k - 1} \right)$
Now, opening the brackets and simplifying the equation, we get,
$ \Rightarrow 6k + 6 = 2k + 1 + 5k - 1$
$ \Rightarrow 6k + 6 = 7k$
[Since, $ - 1 + 1 = 0$]
Now, subtracting both sides by $6k$, we get,
$ \Rightarrow 6 = 7k - 6k$
$ \Rightarrow 6 = k$
Now, changing the sides, we get,
$ \Rightarrow k = 6$
Therefore, $k = 6$, the three consecutive terms form an Arithmetic Progression.
So, the correct answer is “ $k = 6$”.

Note: This problem can also be done using the basic concept of Arithmetic Progression, i.e., the difference between the two consecutive terms of an AP are equal. We could find the difference between the 1st and 2nd term and the difference between the 2nd and 3rd term, then by equating the differences of the consecutive terms, we could get the required result. The formula of arithmetic mean of three terms in AP is also derived from the same concept of AP.