Question

If $2K,\,3K - 1,\,8$ are in A.P, then what is the value of $K$?

Answer 1

It is given that the terms are in A.P. (Arithmetic progression). The progression of the form $a, a + d, a + 2d, a + 3d …$is known as an arithmetic progression where, $a$= first term, and $d$= common difference between the number next to each other.

If three terms $a,b,c$ are in A.P. Then the Arithmetic mean(A.M.) will be the middle term which value will be equal to

$b= \dfrac{a+c}{2}$.
Using this formula we can get the required answer.

Complete step by step solution:
Given: $2K,\,3K - 1,\,8$ are in A.P.

As we know, if three terms $a,b,c$ are in A.P. Their A.M. will be $b=\dfrac{a+c}{2}$

Here, $a=2K,b=3K - 1$ and $c=8$

With the help of the formula let’s find out the value of $K$

$\Rightarrow b = \dfrac{{a + c}}{2}$
$\Rightarrow 3K - 1 = \dfrac{{2K + 8}}{2}$
$\Rightarrow 6K - 2 = 2K + 8$
$\Rightarrow 6K - 2K = 8 + 2$
$\Rightarrow 4K = 10$
$\Rightarrow K = \dfrac{{10}}{4}$
$\Rightarrow K = \dfrac{5}{2}$

So, the value of $K = 2.5$

Note:
Even if you don't remember the formula of arithmetic mean, we can solve this problem by using the logic in Arithmetic progression: $a+0d, a+1d, a+2d$, with common difference $d$ between the terms.

The difference between second term and first term = The difference between the third term and the second term

Applying this for the given terms,

$(3K-1) - (2K) = (8)-(3K-1)$

$\Rightarrow K-1 = 9-3K$

Solving for $K$

$4K = 10$

$\Rightarrow K=\dfrac{10}{4}$.

$\therefore K=5.2$ This is correct as it is matching with the previous $K$ value.