To find a missing term in a geometric sequence, what terms are involved in the computation? If it is an arithmetic sequence, what terms are involved? How many options for the missing term do you have in each case?
See solution.
Practice makes perfect
Let's consider a geometric sequence with one missing term.
3, 6, ?, 24, 48
To find this missing term, we calculate the geometric mean of the second and fourth terms.
Geometric mean:
sqrt(6* 24) = sqrt(144) = 12
Now, the third term could be either 12 or -12. Since the other terms are positive, we discard the negative option. Let's do the same process, but using an arithmetic sequence.
2, 5, 8, ?, 14
To find the missing term, we find the arithmetic mean of the third and fifth terms.
Arithmetic mean:
8+14/2 = 22/2 = 11
Similarities and Differences
Both processes are similar in the sense that, to find a missing term, it is required to perform a certain operation that involves the previous and the next term.
..., a_(n-1), ?, a_(n+1), ...
Geometric Mean
Arithmetic Mean
sqrt(a_(n-1)* a_(n+1)) or -sqrt(a_(n-1)* a_(n+1))
a_(n-1)+a_(n+1)/2
The difference between the two processes — besides the operation required — is that for the geometric sequence there are two options for the missing term (the positive and the negative geometric mean) while for the arithmetic sequence there is only one option (the arithmetic mean).
