Key Information Extraction

The given sequence is: 1, x, x, x, y, y, 9, 16, 18. It is in increasing order.

The number of elements in the sequence is 9.

We are given that Mean = Median = 2 * Mode.

We need to find the value of y.

Determining Mode and Median

Since the sequence is increasing, we have 1 < x < y < 9.

• Mode: The mode is the most frequent number. In the sequence, 'x' appears 3 times and 'y' appears 2 times. Therefore, the mode is x.
• Median: For a sequence of 9 elements, the median is the 5th element. The 5th element is y.

Applying the Mean-Median-Mode Relationship

We are given that Median = 2 * Mode.

Substituting the values:

$y = 2 * x$ (Equation 1)

Calculating the Mean

The sum of the numbers is: $1 + x + x + x + y + y + 9 + 16 + 18 = 3x + 2y + 44$.

The mean is the sum divided by the count (9):

Mean = $ \frac{3x + 2y + 44}{9} $

Equating Mean and Median

We are given that Mean = Median.

Substituting the expressions:

$ \frac{3x + 2y + 44}{9} = y $

Multiply both sides by 9:

$ 3x + 2y + 44 = 9y $

Rearrange the terms:

$ 3x + 44 = 7y $ (Equation 2)

Solving for x and y

Substitute Equation 1 ($y = 2x$) into Equation 2:

$ 3x + 44 = 7(2x) $

$ 3x + 44 = 14x $

Subtract $3x$ from both sides:

$ 44 = 11x $

Solve for $x$:

$ x = \frac{44}{11} $

$ x = 4 $

Now substitute the value of $x$ back into Equation 1 ($y = 2x$) to find $y$:

$ y = 2 \times 4 $

$ y = 8 $

Verification

With x=4 and y=8, the sequence is 1, 4, 4, 4, 8, 8, 9, 16, 18.

• Mode = 4
• Median = 8
• Mean = $ \frac{1+4+4+4+8+8+9+16+18}{9} = \frac{72}{9} = 8 $
• Check: Mean (8) = Median (8) = 2 * Mode (2 * 4 = 8). The conditions are satisfied.

Therefore, the value of y is 8.