As this is posted under geometric sequences, we will proceed under the assumption that the sequences requested must be geometric. A geometric sequence is a sequence whose n^"th" term is given by ar^(n-1) where a is an initial value and r is a common ratio between terms.
By the above, the first 3 terms are a, ar, ar^2, and we are given the conditions that
{(ar = 16), (a+ar+ar^2 = 84):}
Substituting the first condition into the second and subtracting 16, we can rewrite the second condition as
a+ar^2 = 68 or a(r^2+1)=68
Dividing this by ar = 16, we get
(a(r^2+1))/(ar) = 68/16
=> (r^2+1)/r = 17/4
=> r^2-17/4r + 1 = 0
Applying the quadratic formula gives us
r = (17/4+-sqrt((17/4)^2-4))/2 = (17+-15)/8
So r = 4 or r = 1/4
From ar = 16, we also have a = 16/r. Thus, by selecting the different possibilities for r, we get
(a, r) = (4, 4) or (a, r) = (64, 1/4)
Checking our answers, our first 3 terms for (a, r) = (4, 4) would be
4, 16, 64
and our first 3 terms for (a, r) = (64, 1/4) would be
64, 16, 4
Note that in each case, the sum of the terms is 84, as desired.
As a side note, if we do not require a geometric sequence, any sequence a_1, a_2, a_3, ... would suffice so long as the second and third terms satisfy a_2 = 16 and a_3 = 68-a_1.
