Question

The product of three numbers in AP is 224 and the largest number is 7 times the smallest. Find the numbers.

Answer 1

Let the required numbers be (a - d), a and (a + d) .

Then, largest number = (a + d) and smallest number = (a - d).

$\therefore a+d=7(a-d)\Rightarrow 6a=8d\Rightarrow d=\frac{3a}{4}$ . . . (i)

And, $(a-d)\times a\times (a+d)=224$

$\Rightarrow a(a^2-d^2)=224$

$\Rightarrow a(a^2-\frac{9a^2}{16})=224\Rightarrow a^3=(224\times \frac{16}{7})=512=8^3\Rightarrow a=8$

Putting a = 8 in (i), we get $d=(\frac{3}{4}\times 8)=6$

$\therefore$ the required numbers are (8 - 6), 8 and (8 + 6), i.e., 2, 8 and 14.