2003 CEMC Pascal Problems/Problem 2

Problem

The missing number in the geometric sequence $2, 6, 18, 54, \underline{\hspace{2.4em}} , 486$ is

$\text{ (A) }\ 72 \qquad\text{ (B) }\ 90 \qquad\text{ (C) }\ 108 \qquad\text{ (D) }\ 162 \qquad\text{ (E) }\ 216$

Solution 1

We are given in the problem that the sequence of numbers is a geometric series/geometric progression. This means that the common ratio must remain the same.

Using the first and second terms, we can see that the common ratio is $\frac{6}{2} = 3$ . This means that the missing term must be:

$3 \times 54 = \boxed {\textbf {(D) } 162}$

~anabel.disher

Solution 1.1

Similar to solution 1, we can find the common ratio. However, we can also use the sixth term to find the fifth term by dividing the sixth term by the common ratio:

$\frac{486}{3} = \boxed {\textbf {(D) } 162}$

~anabel.disher

Solution 2

We can set up a formula for the $n$ th term of the geometric sequence based on the starting term and the common ratio, and use that to find the $5$ th term of the sequence.