To find an equation for the sequence \(2, 10, 50, \ldots\), we first examine the pattern of the terms.
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Identifying the Terms:
- \(a_1 = 2\)
- \(a_2 = 10\)
- \(a_3 = 50\)
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Finding a Pattern:
- We can observe the relationship between the terms:
- \(10\) is \(2 \times 5\) (or \(2 \times 5^1\))
- \(50\) is \(10 \times 5\) (or \(2 \times 5^2\))
- It appears the terms are \(2\) multiplied by increasing powers of \(5\).
- We can observe the relationship between the terms:
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Defining the General Term:
- The first term can be expressed as:
- \(a_1 = 2 = 2 \times 5^0\)
- \(a_2 = 10 = 2 \times 5^1\)
- \(a_3 = 50 = 2 \times 5^2\)
This suggests that the \(n\)-th term can be represented as: \[ a_n = 2 \times 5^{n-1} \]
- The first term can be expressed as:
Putting it all together, the equation for the \(n\)-th term in the sequence is:
\[ a_n = 2(5)^{n-1} \]
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