Let {ai} with index starting at 0 represent a geometric sequence.
If a7 = 7 and a8 = 28, what is a9?

Given that \(\{a_i\}\) represents a geometric sequence, we know that the ratio between consecutive terms is constant. Let the common ratio be denoted as \(r\).

The terms provided are:
- \(a_7 = 7\)
- \(a_8 = 28\)

In a geometric sequence, we have the relation:
\[
a_{n+1} = a_n \cdot r
\]
Using this, we can express \(a_8\) in terms of \(a_7\):
\[
a_8 = a_7 \cdot r
\]
Substituting the given values:
\[
28 = 7 \cdot r
\]
Solving for \(r\):
\[
r = \frac{28}{7} = 4
\]

Now, to find \(a_9\), we use the fact that:
\[
a_9 = a_8 \cdot r
\]
Substitute \(a_8 = 28\) and \(r = 4\):
\[
a_9 = 28 \cdot 4 = 112
\]

Thus, \(a_9 = 112\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the common ratio in a geometric sequence defined?
2. What is the general formula for the \(n\)-th term of a geometric sequence?
3. How can you determine if a sequence is geometric from its terms?
4. What happens to a geometric sequence if the common ratio is less than 1?
5. How do geometric sequences differ from arithmetic sequences?

**Tip:** In geometric sequences, each term is found by multiplying the previous term by a constant factor (the common ratio).

Let {ai} with index starting at 0 represent a geometric sequence. If a7 = 7 and a8 = 28, what is a9?

This problem explores a geometric sequence where a7 = 7 and a8 = 28. Learn how to determine the common ratio and solve for a9. The solution walks through key algebraic concepts and geometric sequence formulas, making it suitable for students in Grades 8-10.

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