To derive a formula for the sequence \(19, 38, 76, \ldots\), we can first observe the pattern in the terms.
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Identify the first term:
- \( a_1 = 19 \)
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Identify a pattern:
- The second term: \( a_2 = 38 = 19 \times 2 \)
- The third term: \( a_3 = 76 = 19 \times 4 \)
We can observe that each term can be expressed as \( 19 \) multiplied by a power of \( 2 \):
- \( a_1 = 19 \times 2^0 \)
- \( a_2 = 19 \times 2^1 \)
- \( a_3 = 19 \times 2^2 \)
In general, we can express the \( n \)-th term using the power of \( 2 \): \[ a_n = 19 \times 2^{n-1} \]
So, the equation that describes the sequence is: \[ a_n = 19 \times 2^{n-1} \]
Thus, the final answer is: \[ a_n = 19 \cdot 2^{n-1} \]