To solve the problem of finding the missing number in the sequence 2, __ , 18, 54, we will determine if the sequence follows a particular pattern. The sequence appears to be a geometric sequence since each term seems to be connected by a common ratio.
1. Identify the given terms:
- First term ([tex]\(a_1\)[/tex]) = 2
- Third term ([tex]\(a_3\)[/tex]) = 18
- Fourth term ([tex]\(a_4\)[/tex]) = 54
2. Recall that in a geometric sequence, each term is obtained by multiplying the previous term by a common ratio ([tex]\(r\)[/tex]). Therefore, we can express the terms as:
- [tex]\(a_2 = a_1 \cdot r\)[/tex]
- [tex]\(a_3 = a_2 \cdot r = a_1 \cdot r^2\)[/tex]
- [tex]\(a_4 = a_3 \cdot r = a_1 \cdot r^3\)[/tex]
3. Using the given terms:
- Since [tex]\(a_1\)[/tex] is 2 and [tex]\(a_3\)[/tex] is 18, we can set up the following equation using the definition of a geometric sequence:
[tex]\[
a_3 = a_1 \cdot r^2
\][/tex]
Substituting the values we get:
[tex]\[
18 = 2 \cdot r^2
\][/tex]
4. Solve for the common ratio [tex]\(r\)[/tex]:
[tex]\[
r^2 = \frac{18}{2}
\][/tex]
[tex]\[
r^2 = 9
\][/tex]
Taking the square root of both sides:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
5. Determine the missing term ([tex]\(a_2\)[/tex]):
Since [tex]\(a_2 = a_1 \cdot r\)[/tex]:
[tex]\[
a_2 = 2 \cdot 3 = 6
\][/tex]
So, the missing number in the sequence 2, __ , 18, 54 is 6.
