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Home » The sum of three numbers in A.P. is     and the product is 8. Find the numbers.
The sum of three numbers in A.P. is

    \[-3\]

and the product is 8. Find the numbers.
Arithmetic and Geometric Progression | Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal
The sum of three numbers in A.P. is

    \[-3\]

and the product is 8. Find the numbers.

From the question it is given that,

The sum of three numbers in A.P. =

    \[-3\]

The product of three numbers in A.P. =

    \[8\]

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 =

    \[-3\]

    \[\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 \[-35\]

So,

    \[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }d \right)\text{ }\times \text{ }\left( a \right)\text{ }\times \text{ }\left( a\text{ }+\text{ }d \right)\text{ }=\text{ }8 \\ a({{a}^{2}}~\text{ }{{d}^{2}})\text{ }=\text{ }8 \\ -1\text{ }({{\left( -1 \right)}^{2}}~\text{ }{{d}^{2}}~=\text{ }8 \\ 1\text{ }\text{ }{{d}^{2}}~=\text{ }8/-1 \\ 1\text{ }\text{ }{{d}^{2}}~=\text{ }-8 \\ {{d}^{2}}~=\text{ }8\text{ }+\text{ }1 \\ {{d}^{2}}=\text{ }9 \\ d\text{ }=\text{ }\surd 9 \\ d\text{ }=\text{ }\pm 3 \\ \end{array}\]

Therefore, the numbers are if

    \[d\text{ }=\text{ }3\text{ }\left( a\text{ }\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }\text{ }3\text{ }=\text{ }\text{ }4\]

    \[\begin{array}{*{35}{l}} a\text{ }=\text{ }-1 \\ \left( a\text{ }+\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }+\text{ }3\text{ }=\text{ }2 \\ \end{array}\]

If d = – 6

The numbers are

    \[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }d \right)\text{ }=\text{ }\text{ }1\text{ }\text{ }\left( -3 \right)\text{ }=\text{ }\text{ }1\text{ }+\text{ }3\text{ }=\text{ }2 \\ a\text{ }=\text{ }-1 \\ \left( a\text{ }+\text{ }d \right)\text{ }=\text{ }-1\text{ }+\text{ }\left( -3 \right)\text{ }=\text{ }-1\text{ }\text{ }3\text{ }=\text{ }-4 \\ \end{array}\]

Therefore, the numbers

    \[-4,\text{ }-1,\text{ }2,\ldots \text{ }and\text{ }2,\text{ }-1,\text{ }-4,\ldots \]

are in A.P.

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