To derive an equation for the sequence -2, -10, -50, ..., we first need to observe the pattern of the terms.
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Identify the terms:
- \( a_1 = -2 \)
- \( a_2 = -10 \)
- \( a_3 = -50 \)
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Finding the ratio of consecutive terms:
- \(\frac{a_2}{a_1} = \frac{-10}{-2} = 5\)
- \(\frac{a_3}{a_2} = \frac{-50}{-10} = 5\)
This indicates that the terms are generated by multiplying the previous term by 5.
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Identifying the relationship with n:
- Each term can be expressed as a function of its position. It appears that each term can be represented as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 = -2 \) (the first term) and \( r = 5 \) (the common ratio).
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Substituting the values:
- So, substituting the values we have: \[ a_n = -2 \cdot 5^{(n-1)} \]
In conclusion, the equation for the nth term in the sequence is: \[ a_n = -2 \cdot 5^{n-1} \]
So, in the specified format: \[ a_n = -2(5)^{n-1} \]
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