Given that the first term ($a_1$) of the geometric sequence is 1 and the tenth term ($a_{10}$) is 4, we are asked to find the seventeenth term ($a_{17}$).

Step 1: Write the formula for the nth term of a geometric sequence.

The nth term of a geometric sequence is given by:

$a_n = a_1 \times r^{n-1}$

Where:

• $a_n$ is the nth term,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

Step 2: Set up equations using the given information.

From the problem:

• $a_1 = 1$
• $a_{10} = 4$

Using the formula for the nth term for $a_{10}$:

$a_{10} = a_1 \times r^{9}$

Substitute $a_{10} = 4$ and $a_1 = 1$:

$4 = 1 \times r^9$

Thus,

$r^9 = 4$

Step 3: Solve for the common ratio $r$.

To find $r$, take the 9th root of both sides:

$r = \sqrt[9]{4}$

Step 4: Find the 17th term ($a_{17}$).

Now, use the formula for the nth term to find $a_{17}$:

$a_{17} = a_1 \times r^{16}$

Substituting $a_1 = 1$ and $r = \sqrt[9]{4}$:

$a_{17} = 1 \times \left(\sqrt[9]{4}\right)^{16}$

Simplifying further:

$a_{17} = 4^{\frac{16}{9}}$

Now, let's calculate $4^{\frac{16}{9}}$.The seventeenth term of the geometric sequence is approximately $11.76$.

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:

1. How do you find the common ratio of a geometric sequence if two non-consecutive terms are given?
2. What is the general formula for the sum of the first $n$ terms in a geometric sequence?
3. How can you determine if a sequence is geometric given a list of terms?
4. How does the growth rate of a geometric sequence compare to an arithmetic sequence?
5. How would you solve for the nth term if the sequence included negative or fractional common ratios?

Tip: In geometric sequences, knowing just two non-consecutive terms allows you to find the common ratio, which can be applied to determine any other term in the sequence.