To find an equation that describes the sequence \(1, -5, 25, \ldots\), we need to first identify the pattern.
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Finding the pattern:
- The first term \(a_1\) is \(1\).
- The second term \(a_2\) is \(-5\).
- The third term \(a_3\) is \(25\).
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Looking at the terms:
- From \(a_1\) to \(a_2\): \(1 \to -5\) (decrease by \(6\))
- From \(a_2\) to \(a_3\): \(-5 \to 25\) (increase by \(30\))
It appears that the terms are alternating in sign and changing in a non-linear way. A quadratic pattern can often encapsulate such sequences.
- Identifying the quadratic function: Let's assume the form of the \(n\)th term, \(a_n\), is quadratic:
\[ a_n = An^2 + Bn + C \]
We can substitute \(n = 1, 2, 3\) to form equations:
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For \(n = 1\) (\(a_1 = 1\)): \[ A(1)^2 + B(1) + C = 1 \implies A + B + C = 1 \]
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For \(n = 2\) (\(a_2 = -5\)): \[ A(2)^2 + B(2) + C = -5 \implies 4A + 2B + C = -5 \]
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For \(n = 3\) (\(a_3 = 25\)): \[ A(3)^2 + B(3) + C = 25 \implies 9A + 3B + C = 25 \]
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Setting up the system of equations: We now have the following system of equations:
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\(A + B + C = 1\)
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\(4A + 2B + C = -5\)
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\(9A + 3B + C = 25\)
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Subtracting equations: Now, we'll eliminate \(C\) by subtracting the first equation from the others:
From 2 - 1: \[ (4A + 2B + C) - (A + B + C) = -5 - 1 \] \[ 3A + B = -6 \quad \text{(Equation 4)} \]
From 3 - 1: \[ (9A + 3B + C) - (A + B + C) = 25 - 1 \] \[ 8A + 2B = 24 \quad \text{(Equation 5)} \] which simplifies to: \[ 4A + B = 12 \quad \text{(Equation 6)} \]
- Solving the equations: Now, we will solve for \(A\) and \(B\) using equations 4 and 6: From Equation 4: \[ B = -6 - 3A \]
Substitute \(B\) into Equation 6: \[ 4A + (-6 - 3A) = 12 \] \[ 4A - 6 - 3A = 12 \] \[ A - 6 = 12 \] \[ A = 18 \]
Substituting \(A\) back to find \(B\): \[ B = -6 - 3(18) = -6 - 54 = -60 \]
Now use \(A\) and \(B\) to find \(C\): \[ A + B + C = 1 \implies 18 - 60 + C = 1 \] \[ C = 1 + 60 - 18 = 43 \]
- Final equation: The final form of the equation for the sequence is: \[ a_n = 18n^2 - 60n + 43 \]
So, the equation that describes the sequence is:
\[ \boxed{a_n = 18n^2 - 60n + 43} \]
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