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

Step 1: Represent the numbers in A.P.

Let the three numbers in an Arithmetic Progression (A.P.) be represented as $a - d$, $a$, and $a + d$. Here, $a$ is the middle term and $d$ is the common difference.

Step 2: Formulate equations from given conditions

We are given two conditions: first, the product of the three numbers is 224. This gives us the equation $(a - d)a(a + d) = 224$. Second, the largest number ($a + d$) is 7 times the smallest number ($a - d$), which gives us $a + d = 7(a - d)$.

Step 3: Solve the second equation for 'a' in terms of 'd'

Let's simplify the second equation: $a + d = 7(a - d)$. Distribute the 7 on the right side to get $a + d = 7a - 7d$. Rearranging the terms, we get $8d = 6a$, which simplifies to $a = \frac{4}{3}d$.

Step 4: Substitute 'a' into the first equation and solve for 'd'

Now, substitute $a = \frac{4}{3}d$ into the product equation $(a - d)a(a + d) = 224$. This simplifies to $(\frac{1}{3}d)(\frac{4}{3}d)(\frac{7}{3}d) = 224$. Multiplying these terms gives $\frac{28}{27}d^3 = 224$. Solving for $d^3$, we get $d^3 = 216$. Taking the cube root, we find $d = 6$.

Step 5: Find the value of 'a'

With $d = 6$, we can find the value of $a$ using the relation $a = \frac{4}{3}d$. Substituting $d=6$, we get $a = \frac{4}{3}(6) = 8$.

Step 6: Determine the three numbers

Finally, substitute the values of $a=8$ and $d=6$ back into our expressions for the three numbers: $a - d$, $a$, and $a + d$. This gives us $8 - 6 = 2$, $8$, and $8 + 6 = 14$. So the three numbers are 2, 8, and 14.

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