Question

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

Answer 1

Hint: In this question let the three numbers in $A.P.$ be $a - d\;,\;a\;,\;a + d$ and equate the product of them with $224$. Use this to find the values of $d$ (common difference) and $a$ (first term) in order to find the numbers.

Complete step-by-step solution -
According to the question , $224$ is the product of three numbers in arithmetic progression is given,
Let $3$ numbers in $A.P.$ be $a - d\;,\;a\;,\;a + d$
Now $\left( {a - d} \right)a\left( {a + d} \right) = 224$
$ \Rightarrow a\left( {{a^2} - {d^2}} \right) = 224......\left( i \right)$
Now, since the largest number is $7$ times the smallest , i.e. $a + d = 7\left( {a - d} \right)$
Therefore, $d = \dfrac{{3a}}{4}$
Substituting this value of $d$ in equation $\left( i \right)$, we get
$
   \Rightarrow a\left( {{a^2} - \dfrac{{9{a^2}}}{{16}}} \right) = 224 \\
   \Rightarrow a \dfrac{7{a^2}}{16} = 224 \\
   \Rightarrow {a^3} = 512 \\
   \Rightarrow a = 8 \\
   \Rightarrow d = \dfrac{{3a}}{4} = \dfrac{{3 \times 8}}{4} = 6 \\
$
Hence , the three numbers are $2,8,14.$

Note: In such types of questions the concept of arithmetic progression or sequence is used i.e. a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term.