To find an equation for the sequence \( 3, -15, 75, \ldots \), we need to identify the pattern in the sequence.
First, let's observe the terms in relation to their positions:
- \( a_1 = 3 \)
- \( a_2 = -15 \)
- \( a_3 = 75 \)
Calculating the ratios of consecutive terms:
- From \( 3 \) to \( -15 \): \( \frac{-15}{3} = -5 \) (the factor is \(-5\))
- From \( -15 \) to \( 75 \): \( \frac{75}{-15} = -5 \) (the factor is \(-5\))
Based on this pattern, it appears that each term is obtained from the previous term by multiplying by \(-5\).
We can denote the general term by the formula:
\[ a_n = 3 \cdot (-5)^{n-1} \]
where \( n \) is the position in the sequence. Here's the breakdown:
- For \( n = 1 \): \( a_1 = 3 \cdot (-5)^{1-1} = 3 \cdot 1 = 3 \)
- For \( n = 2 \): \( a_2 = 3 \cdot (-5)^{2-1} = 3 \cdot (-5) = -15 \)
- For \( n = 3 \): \( a_3 = 3 \cdot (-5)^{3-1} = 3 \cdot 25 = 75 \)
This accurately generates the sequence.
Thus, the formula for the nth term of the sequence is:
\[ a_n = 3 \cdot (-5)^{n-1} \]