the 65th term in a geometric sequence is 8000, and the 69th term is 500. find the 64th term
ar^64 = 8000
ar^68 = 500
divide the 2nd by the 1st
r^4 = 500/8000 = 1/16
r = ± 1/2
so term 64 = term 65 / r
= 8000/(±1/2) = ± 16000
To find the 64th term of a geometric sequence, we need to first determine the common ratio.
The formula for the nth term of a geometric sequence is:
Tn = a * r^(n-1)
where Tn represents the nth term, a represents the first term of the sequence, r represents the common ratio, and n is the position of the term.
We are given the 65th term, which is 8000, and the 69th term, which is 500. Let's use this information to find the common ratio.
First, substitute the values into the formula:
8000 = a * r^(65-1) ...(1)
500 = a * r^(69-1) ...(2)
Now, divide equation (1) by equation (2):
8000 / 500 = (a * r^(65-1)) / (a * r^(69-1))
16 = (r^64) / (r^68)
Simplify the equation by subtracting the exponent on the right side:
16 = 1 / r^4
To isolate r, we take the reciprocal of both sides:
1/16 = r^4
Now, take the fourth root of both sides of the equation to solve for r:
∛(1/16) = ∛(r^4)
∛(1/16) = r
Simplify the cube root of 1/16:
1/2 = r
Now that we have the common ratio (r = 1/2), we can find the 64th term using the formula:
T64 = a * (1/2)^(64-1)
However, we still need the value of the first term, 'a', to solve for the desired term. Unfortunately, the available information does not provide the value of the first term.
Hence, without the value of the first term, we cannot accurately determine the 64th term of the geometric sequence.