To find the value of [tex]\( x \)[/tex] in the sequence [tex]\( 9, 10, 12, x, 20, 25 \)[/tex], given that the median is 14, let's follow these steps:
1. Understanding the Median:
- The median of a sequence is the middle value when the sequence is ordered.
- If the number of elements in the sequence is even, the median is the average of the two middle numbers.
2. Number of Elements:
- Here, the sequence has 6 elements: [tex]\( 9, 10, 12, x, 20, 25 \)[/tex].
3. Locate the Middle Positions:
- For a sequence with 6 elements, the median will be the average of the 3rd and 4th numbers (when arranged in ascending order).
4. Given Median Value:
- The given median is 14.
- Thus, the average of the 3rd and 4th elements in the sequence must be equal to 14.
5. Set Up the Equation:
- The 3rd and 4th elements in ascending order are 12 and [tex]\( x \)[/tex], respectively.
- We need the average of 12 and [tex]\( x \)[/tex] to be equal to 14:
[tex]\[
\frac{12 + x}{2} = 14
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
- Multiply both sides of the equation by 2 to eliminate the fraction:
[tex]\[
12 + x = 28
\][/tex]
- Subtract 12 from both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 28 - 12
\][/tex]
[tex]\[
x = 16
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 16 \)[/tex].
