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Question
Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
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Solution
\[\text { Let the three numbers be } a - d, a, a + d . \]
\[\text {Their sum } = 27\]
\[ \Rightarrow a - d + a + a + d = 27\]
\[ \Rightarrow 3a = 27\]
\[ \Rightarrow a = 9 . . . (i)\]
\[\text { Product } = (a - d)a(a + d) = 648\]
\[ \Rightarrow a( a^2 - d^2 ) = 648\]
\[ \Rightarrow 9(81 - d^2 ) = 648\]
\[ \Rightarrow (81 - d^2 ) = 72\]
\[ \Rightarrow d^2 = 9\]
\[ \Rightarrow d = \pm 3\]
\[\text { When a = 9, d = 3, we have:} \]
\[6, 9, 12\]
\[\text { When a = 9, d = - 3, we have: } \]
\[12, 9, 6\]
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