From the question it is given that,
The sum of three numbers in A.P. =
![\[-3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4adc8c45b73d97c997db1b8d065f32a8_l3.png "Rendered by QuickLaTeX.com")
The product of three numbers in A.P. =
![\[8\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ba6e07885b354f6c76c81fbd23c392a_l3.png "Rendered by QuickLaTeX.com")
Let us assume the 3 numbers which are in A.P. are, a – d, a, a + d
Now adding 3 numbers = a – d + a + a + d =
![\[\begin{array}{*{35}{l}} 3a\text{ }=\text{ }-3 \\ a\text{ }=\text{ }-3/3 \\ a\text{ }=\text{ }-1 \\ \end{array}\ From the question, product of 3 numbers is \[-35\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-906b471b42fb5cbe15002d4afdd7660c_l3.png "Rendered by QuickLaTeX.com")
So,
![\[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }d \right)\text{ }\times \text{ }\left( a \right)\text{ }\times \text{ }\left( a\text{ }+\text{ }d \right)\text{ }=\text{ }8 \\ a({{a}^{2}}~\text{ }{{d}^{2}})\text{ }=\text{ }8 \\ -1\text{ }({{\left( -1 \right)}^{2}}~\text{ }{{d}^{2}}~=\text{ }8 \\ 1\text{ }\text{ }{{d}^{2}}~=\text{ }8/-1 \\ 1\text{ }\text{ }{{d}^{2}}~=\text{ }-8 \\ {{d}^{2}}~=\text{ }8\text{ }+\text{ }1 \\ {{d}^{2}}=\text{ }9 \\ d\text{ }=\text{ }\surd 9 \\ d\text{ }=\text{ }\pm 3 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7134f261e28ceb68e57076f89e073aa4_l3.png "Rendered by QuickLaTeX.com")
Therefore, the numbers are if
![\[d\text{ }=\text{ }3\text{ }\left( a\text{ }\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }\text{ }3\text{ }=\text{ }\text{ }4\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-58cb20eace9c82ed53b71cc0942b7bc6_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{array}{*{35}{l}} a\text{ }=\text{ }-1 \\ \left( a\text{ }+\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }+\text{ }3\text{ }=\text{ }2 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-3d93a1b995b6c695e7c33466793b8cfb_l3.png "Rendered by QuickLaTeX.com")
If d = – 6
The numbers are
![\[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }\text{ }\left( -3 \right)\text{ }=\text{ }\text{ }1\text{ }+\text{ }3\text{ }=\text{ }2 \\ a\text{ }=\text{ }-1 \\ \left( a\text{ }+\text{ }d \right)\text{ }=\text{ }-1\text{ }+\text{ }\left( -3 \right)\text{ }=\text{ }-1\text{ }\text{ }3\text{ }=\text{ }-4 \\ \end{array}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-448b26303f6318a072f2fb820da88aa2_l3.png "Rendered by QuickLaTeX.com")
Therefore, the numbers
![\[-4,\text{ }-1,\text{ }2,\ldots \text{ }and\text{ }2,\text{ }-1,\text{ }-4,\ldots \]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-958628d4752f6ab5426a4ea6e1401f73_l3.png "Rendered by QuickLaTeX.com")
are in A.P.