If 2, a, b, −54 forms a geometric sequence, find the values of a and b.

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].