Question

If the positive integers $a$, $b$, $c$, $d$ are in AP, then the numbers $abc$, $abd$, $acd$, $bcd$ are in

• A. HP
• B. AP
• C. GP
• D. None of the above

Answer 1

The correct option is A HP

Given numbers are $abc,abd,acd,bcd$

$\Rightarrow \frac{1}{abc}=\frac{d}{abcd}$

$\Rightarrow \frac{1}{abd}=\frac{c}{abcd}$

$\Rightarrow \frac{1}{acd}=\frac{b}{abcd}$

$\Rightarrow \frac{1}{bcd}=\frac{a}{abcd}$

Reciprocals of $abc,abd,acd,bcd$ are $\frac{d}{abcd},\frac{c}{abcd},\frac{b}{abcd},\frac{a}{abcd}$ respectively

Let $k=\frac{1}{abcd}$

Reciprocals of each are $dk,ck,bk,ak$

we know that $a,b,c,d$ are in AP.

So, $ak,bk,ck,dk$ should also in AP

So, $dk,ck,bk,ak$ should also in AP

Since reciprocals are in AP., they are in HP.