Question

Let the positive numbers a, b, c, d be in AP. Then abc, abd, acd, bcd are (a) not in AP/ GP/ HP (b) in AP (c) in GP (d) in HP

Answer 1

Answer:

(d) They are in harmonic progression

Step-by-step explanation:

For reference, give a name to the common difference in the original AP:

Let n = b - a = c - b = d - c.

For a nontrivial AP, we must have n ≠ 0, so a, b, c, d are different numbers.

We can rule out AP and GP quickly.

If AP, then abd - abc = acd - abd  =>  b(d - c) = d(c - b)  =>  bn = dn  => b = d.  This contradiction says it can't be AP.

If GP, then abd / abc = acd / abd  =>  bd = c²  =>  (c - n)(c + n) = c²  =>  c² - n² = c²  =>  n = 0.  Again, a contradiction that says it can't be GP.

To see that it is HP, we need to check that

1 / abd  -  1 / abc  =  1 / acd  -  1 / abd  =  1 / bcd  -  1 / acd.

Multiplying through by abcd, this just means that we need

c - d = b - c = a - b

and this is indeed true (these are all equal to -n).  So they are in HP.

Answer 2

Hello,Buddy!!

Refer The Attachment ⤴️

• abc,abd,acd,bcd are in $\bold{Harmonic\; Progression}$

$\boxed{\tt{@MrSovereign♡}}$

Hope This Helps!!