Answer:
The values are a=-6 and b=18.
Step-by-step explanation:
A Geometric sequence is given by the formula [tex]a_n=mr^{n-1}[/tex] for all [tex]n\geq 1[/tex], where r ≠ 0 is the common ratio and [tex]m[/tex] is the first term of the sequence .
In this problem we know that m= 2, [tex]a=a_{2} [/tex] is the second term of the sequence and [tex]a_{3} =b[/tex] is the third term.
First we need to find the general form of the sequence, we can use the fourth term [tex]a_{4}=-54[/tex] and the value of [tex]m=2[/tex] to find r.
We replace [tex]a_{4}=-54[/tex] and [tex]m=2[/tex] in the formula [tex]a_n=mr^{n-1}[/tex], then we have
[tex]a_4=mr^{4-1}[/tex]
[tex]-54=2r^{4-1}[/tex]
[tex]\frac{-54}{2} =r^3[/tex]
[tex]-27=r^3[/tex]
[tex]r=-3[/tex].
Therefore the general form of the sequence is [tex]a_n=2(-3)^{n-1}[/tex] for all [tex]n \geq 1[/tex].
To find the value of a, we replace n=2 in our formula, so
[tex]a=a_2=2(-3)^{2-1}=2(-3)=-6[/tex].
To find the value of b, we replace n=3 in our formula, so
[tex]b=a_3=2(-3)^{3-1}=2(-3)^{2} =2(9)=18[/tex].
