Question

Let the positive numbers $a$, $b$, $c$, $d$ be in A.P. Then $abc$, $abd$, $acd$, $bcd$ are

• A. NOT in A.P./G.P./H.P.
• B. in A.P.
• C. in G.P.
• D. in H.P.

Answer 1

The correct option is D

in H.P.

Explanation for the correct answer.

Let $A=\left\{1,2,3,4\right\}$ then the positive numbers $a$, $b$, $c$, $d$ be in A.P.

Then the values of

$$\begin{gathered} abc=1\times 2\times 3=6 \\ abd=1\times 2\times 4=8 \\ acd=1\times 3\times 4=12 \\ bcd=2\times 3\times 4=24 \end{gathered}$$

Now that sequence is obtained as: $6,8,12,24,...$

The common difference between each term is

$$\begin{gathered} \Rightarrow 8-6=2 \\ \Rightarrow 12-8=4 \\ \Rightarrow 24-12=6 \end{gathered}$$

Since the common difference is not the same the obtained sequence is not in A.P.

Now let us compare the common ratio between each term

$\frac{8}{6}\ne \frac{12}{8}\ne \frac{24}{12}$

Since the common ratios are not the same the obtained sequence is not in G.P.

Now let us find the common difference between each term in H.P.

$$\begin{gathered} \Rightarrow \frac{1}{8}-\frac{1}{6}=\frac{-2}{48} \\ \Rightarrow \frac{1}{12}-\frac{1}{8}=\frac{-2}{48} \\ \Rightarrow \frac{1}{24}-\frac{1}{12}=\frac{-2}{48} \end{gathered}$$

Here the common difference is the same.

Therefore, We can say that the obtained numbers are in H.P.

Hence, the correct answer is Option (D).