Question

the product of three number in AP is 224 and the largest number is 7 times the smallest find the number

Answer 1

X×Y×z=224
x=7×X
y=X

7×x+x=224
7x+x=224
8x=224
x=224÷8
x=28
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Answer 2

AnswEr:

Let the three numbers in A.P. be a-d, a+d, where d>0. Clearly, a+d is the largest number and a-d is the smallest number.

It is given that :

$\sf \: Product \: of \: numbers = 224 \: \: \: and \: \: \: \\ \sf \: The \: largest \: number = 7 \: \: (the \: smallest \\ \sf \: number)$

$\rightarrow \tt (a - d)a(a + d) = 224 \: \: and \: \: \: \tt \: a + d = 7(a - d) \\ \\ \rightarrow \tt \: a( {a}^{2} + {d}^{2} ) = 224 \: \: and \: \: 6a = 8d \\ \\ \rightarrow \tt \: a( {a}^{2} - {d}^{2} ) = 224 \: \: and \: \: d = \frac{3a}{4} \\ \\ \rightarrow \tt \: a( {a}^{2} - \frac{9}{16} \: {a}^{2} ) = 224 \\ \\ \tt \rightarrow \frac{7 {a}^{3} }{16} = 224 \\ \\ \rightarrow \tt \: {a}^{3} = 512 = {8}^{3} \\ \\ \rightarrow \tt \: a = 8 \\ \\ \rm \: Putting \: a = 8 \: in \: d = \frac{3a}{4} \: \: we \: obtain \\ \rm \: d = 6$

Hence, three numbers are 2,8,14.