Write the first six terms of the geometric sequence with the first​ term, a 1​, and common​ ratio, r.
a 1 equals one half
​, r equals one half
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Part 1
a 2 equals
  
enter your response here  ​(Type an integer or a fraction. Simplify your​ answer.)

To find the terms of a geometric sequence, you can use the formula for the \(n\)th term of a geometric sequence:

\[
a_n = a_1 \times r^{(n-1)}
\]

Given:
- \(a_1 = \frac{1}{2}\)
- \(r = \frac{1}{2}\)

Now, let's find the first six terms:

1. \(a_1 = \frac{1}{2}\)

2. \(a_2 = a_1 \times r = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)

3. \(a_3 = a_1 \times r^2 = \frac{1}{2} \times \left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\)

4. \(a_4 = a_1 \times r^3 = \frac{1}{2} \times \left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{8} = \frac{1}{16}\)

5. \(a_5 = a_1 \times r^4 = \frac{1}{2} \times \left(\frac{1}{2}\right)^4 = \frac{1}{2} \times \frac{1}{16} = \frac{1}{32}\)

6. \(a_6 = a_1 \times r^5 = \frac{1}{2} \times \left(\frac{1}{2}\right)^5 = \frac{1}{2} \times \frac{1}{32} = \frac{1}{64}\)

So, the first six terms of the geometric sequence are:

\[
\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}
\]

Would you like more details or have any questions?

Here are five related questions:

1. What is the 10th term of the sequence?
2. How do you find the sum of the first \(n\) terms of a geometric sequence?
3. If the common ratio were negative, how would the sequence change?
4. How does the geometric sequence differ from an arithmetic sequence?
5. What is the sum of an infinite geometric series with \( |r| < 1 \)?

**Tip:** In a geometric sequence, if the common ratio \(r\) is between -1 and 1 (excluding 0), the terms will get closer and closer to 0 as the sequence progresses.

Write the first six terms of the geometric sequence with the first term, a1, and common ratio, r. Given a1 = 1/2 and r = 1/2.

Learn how to determine the first six terms of a geometric sequence with a given first term of 1/2 and a common ratio of 1/2. This solution provides a step-by-step approach using the geometric sequence formula. Perfect for students in Grades 8-10.

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