Question

If the product of three consecutive numbers in A.P. is $224$ and the largest number is $7$ times the smallest, then which of the following is (are) CORRECT?

• A. The smallest number is $4$.
• B. The sum of all the three numbers is $10$.
• C. The smallest number is $2$.
• D. The sum of all the three numbers is $24$.

Answer 1

Answer: (C), (D)

The correct options are
C The smallest number is $2$.
D The sum of all the three numbers is $24$.

Let the three numbers in A.P. be $(a-d),a,(a+d)$
Given: $(a-d)a(a+d)=224$
$\Rightarrow a(a^2-d^2)=224\cdots \left(1\right)$

Case $:1$ When $(a+d)$ is largest
$a+d=7(a-d)\Rightarrow 6a=8d\Rightarrow d=\frac{3a}{4}\cdots \left(2\right)$
Using equation $\left(1\right)$,
$a(a^2-\frac{9a^2}{16})=224\Rightarrow 7a^3=224\times 16\Rightarrow a^3=32\times 16=2^9\Rightarrow a=2^3=8$
Using equation $\left(2\right),$
$d=\frac{3\times 8}{4}=6$
Therefore, the required A.P. is $2,8,14$

Case $:2$ When $(a-d)$ is largest
$a-d=7(a+d)\Rightarrow -6a=8d\Rightarrow d=-\frac{3a}{4}\cdots \left(3\right)$
Using equation $\left(1\right)$,
$a(a^2-\frac{9a^2}{16})=224\Rightarrow 7a^3=224\times 16\Rightarrow a^3=32\times 16=2^9\Rightarrow a=2^3=8$
Using equation $\left(3\right),$
$d=-\frac{3\times 8}{4}=-6$
Therefore, the required A.P. is $14,8,2$

Hence, the smallest is $2$ and sum of all three numbers is $24$.