The 5th term in a geometric sequence is 720. The 7th term is 20. What are the possible values of the 6th term of the sequence?
Let the common ratio of the geometric sequence be denoted by r, and let the first term be denoted by a.
Given that the 5th term is 720, we have:
a * r^4 = 720
Given that the 7th term is 20, we have:
a * r^6 = 20
Dividing the equation for the 5th term by the equation for the 7th term gives:
r^2 = 36
r = ±6
If r = 6:
a * 6^4 = 720
a * 1296 = 720
a = 720 / 1296
a = 5/9
Therefore, the 6th term would be:
a * r^5 = (5/9) * 6^5 = 5/9 * 7776 = 4320
If r = -6:
a * (-6)^4 = 720
a * 1296 = 720
a = 720 / 1296
a = 5/9
Therefore, the 6th term would be:
a * (-6)^5 = (5/9) * (-6)^5 = 5/9 * -7776 = -4320
Therefore, the possible values of the 6th term are 4320 and -4320.