We want to find the missing terms of the given geometric sequence.
2, , , , 162, ...
We will start by calculating the geometric mean between the first and the fifth term. To find the geometric mean between two numbers, we calculate the square root of their product.
Geometric mean=sqrt(2 * 162)Let's simplify the right-hand side of this equation!
We found that the geometric mean between 2 and 162 is 18. Let's use this to calculate the possible values of the third term of the given sequence.
If the common ratio is positive, every term of the sequence has the same sign. Since the first term ( 2) is positive, every term of the sequence would be positive.
If the common ratio is negative, the signs of the terms alternate. Since the first term ( 2) is positive, the second term would be negative and the third term would be positive.
We do not know the common ratio, but both of these cases tell us that the third term is positive. Therefore, we can conclude that it equals 18. Let's use this to find the second and fourth terms by calculating two more geometric means. Recall that by its definition, a geometric mean is always positive. However, the value of a term can be either positive or negative.
Geometric mean=± sqrt(a * b)
Second term=±sqrt(2 * 18)
Fourth term=±sqrt(18 * 162)
Second term=±sqrt(36)
Fourth term=±sqrt(2916)
Second term=± 6
Fourth term=± 54
We have two possible values for the second and fourth terms. Finally, let's complete the given sequence.
ccccc
2 & 6 & 18 & 54 & 162 [0.8em]
& & or & & [0.8em]
2 & -6 & 18 & -54 & 162
In short, the missing terms are ±6, 18, and ± 54.
