Sorry the full question is: If the third term of a of a geometric sequence is 7, and the 7th term is 16.807, what is the common ratio?
ar^3 = 7
ar^7 = 16807
So, that means that
ar^7/ar^3 = r^4 = 16807/7
Now it's easy to find r.
What is the r ?
To find the common ratio of a geometric sequence, you can use the formula:
\[\text{{term}}_n = \text{{term}}_1 \cdot r^{(n-1)}\]
where \(\text{{term}}_n\) represents the nth term, \(\text{{term}}_1\) represents the first term, \(r\) represents the common ratio, and \(n\) represents the position of the term.
Given that the third term of the geometric sequence is 7, you can substitute the values into the formula:
\[7 = \text{{term}}_1 \cdot r^{(3-1)}\]
Simplifying, we get:
\[7 = \text{{term}}_1 \cdot r^2\]
Similarly, for the 7th term, we have:
\[16.807 = \text{{term}}_1 \cdot r^{(7-1)}\]
Simplifying, we get:
\[16.807 = \text{{term}}_1 \cdot r^6\]
We now have a system of two equations:
Equation 1: \(7 = \text{{term}}_1 \cdot r^2\)
Equation 2: \(16.807 = \text{{term}}_1 \cdot r^6\)
To solve this system of equations, divide the two equations:
\[\frac{{16.807}}{{7}} = \frac{{\text{{term}}_1 \cdot r^6}}{{\text{{term}}_1 \cdot r^2}}\]
Simplifying further:
\[\frac{{16.807}}{{7}} = r^4\]
Now you can solve for the common ratio, \(r\), by taking the fourth root of \(\frac{{16.807}}{{7}}\):
\[r = \sqrt[4]{{\frac{{16.807}}{{7}}}}\]
Using a calculator, you can find that \(r \approx 1.391\).
Therefore, the common ratio of the given geometric sequence is approximately 1.391.