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Q.
The product of three numbers in an $A.P$. is $224$, and the largest number is $7$ times the smallest. Find the numbers.
Sequences and Series
Solution:
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 $2, 8,14$.