Answer:
Step-by-step explanation:
You want the first term and the common ratio of the geometric sequence with fourth term 13.5 and sum of first three terms of 74.
General term
The expression for the n-th term of a geometric sequence with first term 'a' and common ratio r is ...
[tex]a_n=a\,r^{n-1}[/tex]
13.5 being the 4th term gives one equation in a, r:
13.5 = a·r³
Sum
The sum of n terms of a geometric sequence is ...
[tex]S_n=a\cdot\dfrac{1-r^n}{1-r}[/tex]
74 being the sum of the first 3 terms gives another equation in a, r:
[tex]74=a\cdot\dfrac{1-r^3}{1-r}[/tex]
Solution
We can solve these equations simultaneously to find a, r. Perhaps the easiest way is to use a graphing calculator to find the point (a, r) where the curves intersect. The attachment shows this is ...
(a, r) = (32, 0.75)
The first term is 32, and the common ratio is 3/4.
__
Additional comment
The second equation can be divided out, and the first equation used to substitute for 'a'. The result can be rearranged to the cubic ...
(148/27)r³ -r² -r -1 = 0
This has one real zero at r = 3/4.
