Question

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

• A. $1, 5, 7$
• B. $2, 6, 8$
• C. $2, 8, 14$
• D. $4, 8, 10$

Answer 1

Answer: (C)

$2, 8, 14$

Answer 2

Let the three numbers be $a - d, a, a + d (d > 0)$ Now $(a- d) \,a(a + d) = 224$ $\Rightarrow a(a^2 - d^2) = 224 \quad . . .(i)$ Also, $a + d = 7 (a- d)$ $\Rightarrow d = \frac{3a}{4}$ Substituting this value of $d$ in $(i)$, we get $a\left(a^{2} - \frac{9a^{2}}{16}\right) = 224$ $\Rightarrow a = 8$ and $d= \frac{3}{4} \times 8 = 6$ Hence, the three numbers are $2, 8,14$.