To determine the general term [tex]\(a_n\)[/tex] of the sequence [tex]\(-10, 50, -250, 1250, -6250, \ldots\)[/tex], we will follow these steps:
1. Identify the first term [tex]\((a_1)\)[/tex]:
The first term of the sequence is [tex]\(a_1 = -10\)[/tex].
2. Determine the common ratio [tex]\((r)\)[/tex]:
To find the common ratio, divide the second term by the first term:
[tex]\[
r = \frac{50}{-10} = -5
\][/tex]
Verify the common ratio with the next terms:
[tex]\[
\frac{-250}{50} = -5 \quad \text{and} \quad \frac{1250}{-250} = -5
\][/tex]
The common ratio [tex]\(r\)[/tex] is consistently [tex]\(-5\)[/tex].
3. Formulate the general term for a geometric sequence:
The general formula for the nth term of a geometric sequence is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
Plugging in the values for [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex]:
[tex]\[
a_n = -10 \cdot (-5)^{(n-1)}
\][/tex]
This provides the simplified general term of the sequence.
Therefore, the general term [tex]\(a_n\)[/tex] of the given sequence is:
[tex]\[
a_n = -10 \cdot (-5)^{(n-1)}
\][/tex]
