Question
Find the 10th term of the geometric sequence 10, comma, minus, 20, comma, 40, comma, dot dot dot10,−20,40,...
Answers
GPT-4o mini
To find the 10th term of the geometric sequence given by the terms: \(10, -20, 40, \ldots\), we first need to determine the common ratio of the sequence.
The common ratio \(r\) can be found by dividing the second term by the first term:
\[
r = \frac{-20}{10} = -2
\]
Next, we can express the \(n\)-th term of a geometric sequence using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.
In this case:
- \(a_1 = 10\),
- \(r = -2\),
- and we want to find \(a_{10}\) (the 10th term).
Substituting these values into the formula gives:
\[
a_{10} = 10 \cdot (-2)^{10-1} = 10 \cdot (-2)^9
\]
Calculating \((-2)^9\):
\[
(-2)^9 = -512
\]
Now, substituting this back into the formula for \(a_{10}\):
\[
a_{10} = 10 \cdot (-512) = -5120
\]
Therefore, the 10th term of the geometric sequence is:
\[
\boxed{-5120}
\]
The common ratio \(r\) can be found by dividing the second term by the first term:
\[
r = \frac{-20}{10} = -2
\]
Next, we can express the \(n\)-th term of a geometric sequence using the formula:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.
In this case:
- \(a_1 = 10\),
- \(r = -2\),
- and we want to find \(a_{10}\) (the 10th term).
Substituting these values into the formula gives:
\[
a_{10} = 10 \cdot (-2)^{10-1} = 10 \cdot (-2)^9
\]
Calculating \((-2)^9\):
\[
(-2)^9 = -512
\]
Now, substituting this back into the formula for \(a_{10}\):
\[
a_{10} = 10 \cdot (-512) = -5120
\]
Therefore, the 10th term of the geometric sequence is:
\[
\boxed{-5120}
\]