To find the 6th term of the given geometric sequence [tex]\(-2, 10, -50, \ldots\)[/tex], we will use the formula for the [tex]\(n\)[/tex]th term of a geometric sequence:
[tex]\[a_n = a_1 \cdot r^{(n-1)}\][/tex]
Step 1: Identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex])
The first term ([tex]\(a_1\)[/tex]) is given as:
[tex]\[a_1 = -2\][/tex]
To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term:
[tex]\[r = \frac{10}{-2} = -5\][/tex]
Step 2: Plug in the values into the formula
We are looking for the 6th term ([tex]\(a_6\)[/tex]), so we will use [tex]\(n = 6\)[/tex]:
[tex]\[a_6 = a_1 \cdot r^{(6-1)}\][/tex]
Substitute [tex]\(a_1 = -2\)[/tex] and [tex]\(r = -5\)[/tex]:
[tex]\[a_6 = -2 \cdot (-5)^{5}\][/tex]
Step 3: Calculate [tex]\((-5)^{5}\)[/tex]
[tex]\[
(-5)^{5} = (-5) \cdot (-5) \cdot (-5) \cdot (-5) \cdot (-5) = -3125
\][/tex]
Step 4: Multiply by the first term
[tex]\[a_6 = -2 \cdot (-3125)\][/tex]
[tex]\[
a_6 = 6250
\][/tex]
Therefore, the 6th term of the sequence is:
[tex]\[ \boxed{6250} \][/tex]
