SOLUTION: The third and tenth terms of a geometric sequence are 9 and 19683 respectively. Which term of the sequence is 59,049?

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Question 233392: The third and tenth terms of a geometric sequence are 9 and 19683 respectively. Which term of the sequence is 59,049?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The general form for geometric sequences is:
a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28%28n-1%29%29
We will use this to solve the problem. We'll start with the 3rd term:
9+=+a%5B1%5D%2Ar%28%283-1%29%29+=+a%5B1%5D%2Ar%5E2
Solving for a%5B1%5D:
9%2Fr%5E2+=+a%5B1%5D

Now we can use this in the equation for the 10th term:
19683+=+%289%2Fr%5E2%29%2Ar%5E%28%2810-1%29%29+=+%289%2Fr%5E2%29%2Ar%5E9+=+9r%5E7
Solving for r...
Divide both sides by 9:
2187+=+r%5E7
Raising both sides to the (1/7) power:
2187%5E%281%2F7%29+=+%28r%5E7%29%5E%281%2F7%29
3+=+r and a%5B1%5D+=+9%2Fr%5E2+=+9%2F3%5E2+=+1

Now that we know a%5B1%5D and r, we can find the answer to the question, which term is 59049?
59049+=+1%2A3%5E%28%28n-1%29%29+=+3%5E%28%28n-1%29%29
So the question is, what power of 3 is 59049? Since n must be a natural number, we could use trial and error to find this. Or we can use logarithms:
log%28%2859049%29%29+=+log%28%283%5E%28%28n-1%29%29%29%29
Using a property of logarithms, log%28a%2C+%28x%5Ey%29%29+=+y%2Alog%28a%2C+%28x%29%29
log%28%2859049%29%29+=+%28n-1%29%2Alog%28%283%29%29
Dividing both sides by log%28%283%29%29:
log%28%2859049%29%29%2Flog%28%283%29%29+=+n-1
Add one to both sides:
log%28%2859049%29%29%2Flog%28%283%29%29+%2B+1+=+n
Using our calculators to find the two logarithms:
10+%2B+1+=+n
11+=+n
So 59049 is the eleventh term in the sequence.