Answer:the first term is 81
Step-by-step explanation:
The formula for determining the nth term of a geometric sequence is expressed as
Tn = ar^(n - 1)
Where
Tn represents the nth term
n represents the number of terms
r represents the common ratio
The 3rd term of the geometrical sequence is larger than the 2nd
term by 36. This means that
T3 - T2 = 36 - - - - - - - - - - 1
The product of these two terms is -243. This means that
T2 × T3 = - 243
T2 = - 243/T3
Substituting T2 = - 243/T3 into equation 1, it becomes
T3 - (- 243/T3) = 36
T3 + 243/T3 = 36
T3^2 + 243 = 36T3
T3^2 - 36T3 + 243 = 0
T3^2 - 27T - 9T3 + 243 = 0
T3(T3 - 27) - 9(T3 - 27) = 0
T3 - 9 = 0 or T3 - 27 = 0
T3 = 9 or T3 = 27
Substituting T3 = 9 or T3 = 27 into
T2 = - 243/T3 = 27, it becomes
T2 = - 243/9 or - 243/27
T2 = - 27 or T2 = - 9
Therefore,
The 3rd term is 9
The 2nd term is - 27
The common ratio, r would be
T3/T2 = 9/-27 = - 1/3
The expression for the third term would be
9 = a × - 1/3^(3 - 1)
9 = a × (- 1/3)^2
9 = a × 1/9
a = 9 × 9 = 81
