if the first and and tenth terms of a geometric sequence are 1 and 4, find the seventeenth term

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.

Learn how to find the 17th term of a geometric sequence where the first term is 1 and the tenth term is 4. This problem involves calculating the common ratio and applying the formula for the nth term of a geometric sequence. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation