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Question
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
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Solution
\[\text { Let the three numbers be } (a - d), a, (a + d) . \]
\[\text { Sum } = 24\]
\[ \Rightarrow (a - d) + a + (a + d) = 24\]
\[ \Rightarrow 3a = 24\]
\[ \Rightarrow a = 8 . . . (i)\]
\[\text { Product } = a(a - d)(a + d) = 440\]
\[ \Rightarrow a( a^2 - d^2 ) = 440\]
\[ \Rightarrow 8(64 - d^2 ) = 440 \left(\text { Form } (i) \right)\]
\[ \Rightarrow (64 - d^2 ) = 55\]
\[ \Rightarrow d^2 = 9\]
\[ \Rightarrow d = \pm 3\]
\[\text { With a = 8, d = 3, we have }: \]
\[5, 8, 11\]
\[\text { With a = 8, d = - 3, we have: } \]
\[11, 8, 5\]
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