Question

the sum of three numbers in an AP is 75 and the product of the greatest and the least is 161. find the number.​

Answer 1

`7,11,15,19,23`

Explanation:

Let the 5 Numbers be:

`a - 2d, a - d, a, a + d, a+ 2d`

The sum of these 5 numbers are:

`( a - 2d ) + ( a - d ) + ( a ) + ( a + d ) + ( a+ 2d ) = 5a`

Given in the statement, we now have

`5a = 75`

⇒
`a = 15`

The product of the greatest and the least numbers of the series

=
`( a - 2d ) ( a + 2d )`

=
`a^(2)` -
`(2d)^(2)`

=
`a^(2) - 4d^(2)`

By the statement

=
`a^(2) - 4d^(2) = 161`

Substituting
`a = 15`

`15^(2) - 4d^(2) = 161`

⇒
`225 - 4d^(2) = 161`

⇒
`4d^(2) =64`

⇒
`d^(2) = 16`

⇒
`d =` ±
`4`

Now, taking both
`+4` and
`-4` we get the following:

`7,11,15,19,23`