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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The general form for geometric sequences is:

We will use this to solve the problem. We'll start with the 3rd term:

Solving for :


Now we can use this in the equation for the 10th term:

Solving for r...
Divide both sides by 9:

Raising both sides to the (1/7) power:

and

Now that we know and r, we can find the answer to the question, which term is 59049?

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:

Using a property of logarithms,

Dividing both sides by :

Add one to both sides:

Using our calculators to find the two logarithms:


So 59049 is the eleventh term in the sequence.