To determine the general term, [tex]\(a_n\)[/tex], of the given sequence [tex]\(-10, 50, -250, 1250, -6250, \ldots\)[/tex], we first identify the type of sequence and its properties.
1. Identify the Pattern: This sequence exhibits alternating signs and grows rapidly in magnitude. Hence, it suggests a geometric sequence with a common ratio.
2. Determine the Common Ratio:
- The first term (denoted [tex]\(a_1\)[/tex]) is [tex]\(-10\)[/tex].
- To find the common ratio ([tex]\(r\)[/tex]), divide the second term by the first term:
[tex]\[
r = \frac{50}{-10} = -5
\][/tex]
3. General Formula for the nth Term of a Geometric Sequence:
- The general formula for the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\(a_1\)[/tex] is the first term and [tex]\(r\)[/tex] is the common ratio.
4. Substitute the Known Values:
- Here, [tex]\(a_1 = -10\)[/tex] and [tex]\(r = -5\)[/tex].
- Therefore, the general term [tex]\(a_n\)[/tex] can be written as:
[tex]\[
a_n = -10 \cdot (-5)^{(n-1)}
\][/tex]
So, the nth term [tex]\(a_n\)[/tex] of the sequence is:
[tex]\[
a_n = -10 \cdot (-5)^{(n-1)}
\][/tex]
