To determine the equation for the sequence 5, -10, 20, we first need to identify the pattern.
-
Identify the terms:
- 1st term (n = 1): \( a_1 = 5 \)
- 2nd term (n = 2): \( a_2 = -10 \)
- 3rd term (n = 3): \( a_3 = 20 \)
-
Look for the pattern: Let's calculate the ratios between the terms:
-
From \( a_1 \) to \( a_2 \): \[ \text{ratio} = \frac{-10}{5} = -2 \]
-
From \( a_2 \) to \( a_3 \): \[ \text{ratio} = \frac{20}{-10} = -2 \]
This indicates that each term is multiplied by -2 to get the next term.
-
General formula: We can express \( a_n \) as follows:
- The first term is \( 5 \).
- Each subsequent term is obtained by multiplying the previous term by -2.
Therefore, we can write the equation for the \( n \)-th term as: \[ a_n = 5 \cdot (-2)^{(n-1)} \]
So, the final answer is: \[ a_n = 5 \cdot (-2)^{(n-1)} \]