Question

-2, 10, -50, 250, ...
what is the 11th term of the geometric sequence?
19,531,250
97,656,250
-19,531,250
-5120

Explanation:

Step1: Identify the geometric sequence parameters

A geometric sequence has the form \( a_n = a_1 \cdot r^{n - 1} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
Given the sequence: \(-2, 10, -50, 250, \dots\)
First term \( a_1 = -2 \).
Common ratio \( r = \frac{10}{-2} = -5 \) (check: \( \frac{-50}{10} = -5 \), \( \frac{250}{-50} = -5 \), so \( r = -5 \)).

Step2: Apply the geometric sequence formula for the 11th term

We need to find \( a_{11} \). Using the formula \( a_n = a_1 \cdot r^{n - 1} \), substitute \( a_1 = -2 \), \( r = -5 \), and \( n = 11 \):
\( a_{11} = -2 \cdot (-5)^{11 - 1} \)
Simplify the exponent: \( 11 - 1 = 10 \), so \( a_{11} = -2 \cdot (-5)^{10} \).

Step3: Calculate \( (-5)^{10} \) and multiply by -2

\( (-5)^{10} = 5^{10} = 9765625 \) (since even exponent makes it positive).
Then, \( a_{11} = -2 \cdot 9765625 = -19531250 \). Wait, but let's check the options. Wait, maybe I made a mistake? Wait, no—wait the options have \(-19,531,250\) (which is \(-19531250\) with commas). Wait, let's recheck the ratio. Wait, \( -2 \times (-5) = 10 \), \( 10 \times (-5) = -50 \), \( -50 \times (-5) = 250 \), so ratio is correct. Then \( n = 11 \), so exponent is \( 10 \). \( (-5)^{10} = 9765625 \), times \(-2\) is \(-19531250\), which matches the option \(-19,531,250\) (commas for thousands separators).

Answer:

\(-19,531,250\) (corresponding to the option with this value)