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#.
