In a geometric sequence,T6=-243 and T3=72
Determine the first term.(2)
To find the first term of the geometric sequence, we can use the formula for finding a term in a geometric sequence:
Tn = a * r^(n-1)
where:
Tn = the nth term
a = the first term
r = the common ratio
n = the term number
We are given that T6 = -243 and T3 = 72.
Using the formula for T6:
-243 = a * r^(6-1)
-243 = a * r^5
Using the formula for T3:
72 = a * r^(3-1)
72 = a * r^2
Now, we can divide these two equations to eliminate the first term 'a':
-243 / 72 = (a * r^5) / (a * r^2)
-3.375 = r^3
Taking the cube root of both sides:
r ā -1.5
Now, we can substitute this value of r back into the equation for T3 to solve for the first term 'a':
72 = a * (-1.5)^2
72 = a * 2.25
a = 72 / 2.25
a = 32
Therefore, the first term of the geometric sequence is 32.