Question
Find the 8th term of the geometric sequence 10, comma, minus, 40, comma, 160, comma, dot dot dot10,−40,160,...
Answers
GPT-4o mini
To find the 8th term of the geometric sequence given by \(10, -40, 160, \ldots\), we first need to identify the first term (denoted as \(a\)) and the common ratio (denoted as \(r\)) of the sequence.
1. **Identify the first term \(a\)**:
\[
a = 10
\]
2. **Identify the common ratio \(r\)**:
We find the ratio by dividing the second term by the first term:
\[
r = \frac{-40}{10} = -4
\]
To verify, let's find the ratio between the third term and the second term:
\[
r = \frac{160}{-40} = -4
\]
The common ratio \(r\) is confirmed to be \(-4\).
3. **Find the 8th term**:
The \(n\)-th term of a geometric sequence can be calculated using the formula:
\[
a_n = a \cdot r^{n-1}
\]
For the 8th term (\(n = 8\)):
\[
a_8 = 10 \cdot (-4)^{8-1} = 10 \cdot (-4)^7
\]
4. **Calculate \((-4)^7\)**:
\[
(-4)^7 = -16384
\]
5. **Calculate \(a_8\)**:
\[
a_8 = 10 \cdot (-16384) = -163840
\]
Thus, the 8th term of the geometric sequence is \(\boxed{-163840}\).
1. **Identify the first term \(a\)**:
\[
a = 10
\]
2. **Identify the common ratio \(r\)**:
We find the ratio by dividing the second term by the first term:
\[
r = \frac{-40}{10} = -4
\]
To verify, let's find the ratio between the third term and the second term:
\[
r = \frac{160}{-40} = -4
\]
The common ratio \(r\) is confirmed to be \(-4\).
3. **Find the 8th term**:
The \(n\)-th term of a geometric sequence can be calculated using the formula:
\[
a_n = a \cdot r^{n-1}
\]
For the 8th term (\(n = 8\)):
\[
a_8 = 10 \cdot (-4)^{8-1} = 10 \cdot (-4)^7
\]
4. **Calculate \((-4)^7\)**:
\[
(-4)^7 = -16384
\]
5. **Calculate \(a_8\)**:
\[
a_8 = 10 \cdot (-16384) = -163840
\]
Thus, the 8th term of the geometric sequence is \(\boxed{-163840}\).
Find the 11th term of the geometric sequence 6, comma, minus, 18, comma, 54, comma, dot dot dot6,−18,54,...