Let the three terms of the arithmetic progression (AP) be \( a - d, a, a + d \), where \( a \) is the middle term and \( d \) is the common difference.
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The sum of the three terms is given as: \[ (a - d) + a + (a + d) = 3a \] Setting this equal to 15, we have: \[ 3a = 15 \implies a = 5 \]
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The product of the three terms is given as: \[ (a - d) \cdot a \cdot (a + d) = 80 \] Using the value of \( a \): \[ (5 - d) \cdot 5 \cdot (5 + d) = 80 \]
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Simplifying the product: \[ 5 (5 - d)(5 + d) = 80 \] Using the difference of squares: \[ (5 - d)(5 + d) = 25 - d^2 \] Thus: \[ 5(25 - d^2) = 80 \] Dividing by 5: \[ 25 - d^2 = 16 \] Rearranging gives: \[ d^2 = 25 - 16 = 9 \] Taking the square root: \[ d = 3 \quad \text{or} \quad d = -3 \]
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Now we can find the three terms of the AP:
If \( d = 3 \): \[ 5 - 3 = 2, \quad 5, \quad 5 + 3 = 8 \] So, the AP is \( 2, 5, 8 \).
If \( d = -3 \): \[ 5 - (-3) = 8, \quad 5, \quad 5 + (-3) = 2 \] The AP would still be the same set of numbers \( 8, 5, 2 \), just in a different order.
Thus, the arithmetic progression is \( 2, 5, 8 \).
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