Directions (55-57): Read the following table carefully and answer the questions given below.

In the table some numbers are given and the probability of picking that number is given.

NUMBERS	Probability
Q	$\frac{1}{6}$
2.5	$\frac{1}{3}$
Q − 2.5	$\frac{1}{3}$
P	$\frac{1}{6}$
$\frac{4Q-5}{2}$	$\frac{1}{6}$
18.75	$\frac{1}{6}$

If the series ‘Q,  2.5,  Q − 2.5,  P,  $\frac{4Q-5}{2}$,  18.75’  follows a certain pattern, then find the value of P.

The correct option is 3.75.

To find the value of $P$, we need to determine the underlying pattern of the given series:
$Q$, $2.5$, $Q-2.5$, $P$, $\frac{4Q-5}{2}$, $18.75$

Let the terms of the series be represented as $T_1$, $T_2$, $T_3$, $T_4$, $T_5$, and $T_6$ respectively. Let us test a multiplication pattern where each successive term is obtained by multiplying the previous term by an increasing decimal factor (starting from 0.5, incrementing by 0.5 at each step):
$T_2=T_1\times 0.5$
$T_3=T_2\times 1.0$
$T_4=T_3\times 1.5$
$T_5=T_4��2.0$
$T_6=T_5\times 2.5$

Step 1: Find the value of $Q$
From the first step of the pattern:
$T_2=T_1\times 0.5$
Given that $T_1=Q$ and $T_2=2.5$:
$2.5=Q\times 0.5$
$Q=\frac{2.5}{0.5}=5$

Step 2: Verify the third term $T_3$
Using $Q=5$, the third term is:
$T_3=Q-2.5=5-2.5=2.5$
According to our pattern:
$T_3=T_2\times 1.0=2.5\times 1.0=2.5$
The pattern holds true.

Step 3: Calculate the value of $P$ (the fourth term $T_4$)
According to our pattern:
$P=T_4=T_3\times 1.5$
$P=2.5\times 1.5=3.75$

Step 4: Verify with the remaining terms
Let us check the fifth term $T_5$:
Using the formula given in the series:
$T_5=\frac{4Q-5}{2}=\frac{4\left(5\right)-5}{2}=\frac{15}{2}=7.5$
Using our pattern:
$T_5=T_4\times 2.0=3.75\times 2=7.5$
This matches perfectly.
Now let us check the sixth term $T_6$:
Using our pattern:
$T_6=T_5\times 2.5=7.5\times 2.5=18.75$
This also matches the given sixth term of the series.

Thus, the value of $P$ is indeed 3.75.