The product of three numbers in A.P. is 224, and the largest number is `7` times the smallest. Find the numbers.

Explanation

Let the required numbers be (a-d) and (a+d) .
Then , largest number = (a+d) and smallest number = ( a-d)
` a+d= 7( a-d) Rightarrow 6a =8d Rightarrow d= (3a)/4`
And ` , (a-d) xx a xx ( a+d) = 224`
` Rightarrow a(a^(2) - (9a^(2))/16) =224 Rightarrow a^(3) = (224 xx 16/7) = 512 =8^(2) Rightarrow a=8`
putting a =8 in (i) , we get d ` = (3/4 xx 8) =6`
the required number are ( 8-6), 8 and ( 8+6) ,i.e, 2,8,and 14