Get detailed and accurate responses to your questions on IDNLearn.com. Ask your questions and get detailed, reliable answers from our community of experienced experts.

Find the eighth term of the geometric sequence, given the first term and the common ratio.

[tex]a_1 = 2 \text{ and } r = -2[/tex]

[tex]\[\boxed{\phantom{answer}}\][/tex]


Sagot :

To find the eighth term of a geometric sequence, we use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:

[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]

Given:
- The first term [tex]\(a_1 = 2\)[/tex]
- The common ratio [tex]\(r = -2\)[/tex]
- We want to find the eighth term, so [tex]\(n = 8\)[/tex]

Let's substitute these values into the formula and solve step-by-step.

1. Start with the formula:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]

2. Substitute the known values:
[tex]\[ a_8 = 2 \cdot (-2)^{(8-1)} \][/tex]

3. Calculate the exponent:
[tex]\[ a_8 = 2 \cdot (-2)^7 \][/tex]

4. Evaluate the power of [tex]\(-2\)[/tex]:
[tex]\[ (-2)^7 = -128 \][/tex]

5. Multiply the first term by this result:
[tex]\[ a_8 = 2 \cdot (-128) \][/tex]

6. Conclude the multiplication:
[tex]\[ a_8 = -256 \][/tex]

Therefore, the eighth term of the geometric sequence is [tex]\(\boxed{-256}\)[/tex].