Animated Solution for Mathematics - Sequence and Series: Let the positive numbers $a,b,c,d$ be in A.P. Then $abc,abd,acd,bcd$ are

Welcome, future engineers! Today, we are going to unravel a beautiful problem that sits at the intersection of algebraic elegance and logical deduction. We are given four positive numbers, $a,b,c,d$, which are in an Arithmetic Progression (A.P.).

Our mission is to determine the nature of the sequence formed by their products taken three at a time: $abc,abd,acd,bcd$. This might look like a daunting algebraic mess, but there is a hidden rhythm here waiting to be discovered.

The Setup

First, let us ground ourselves. We have four numbers in A.P., meaning the difference between consecutive terms is constant.

Instead of getting bogged down in the definition of A.P. immediately, let us look at our target sequence: $abc,abd,acd,bcd$. Notice that each term is a product of three variables, missing exactly one from the set $\{a,b,c,d\}$.

In mathematics, symmetry is rarely a coincidence; it is a signpost pointing toward an elegant solution.

The Transformation

Here is the 'Aha!' moment. We know that if we have a sequence in A.P., we can manipulate it without breaking the progression. Specifically, if we divide every term of an A.P. by the same non-zero constant, the resulting sequence remains in A.P.

Since $a,b,c,d$ are positive, their product $abcd$ is definitely not zero. Let us divide our target sequence by this product, $abcd$:

Look at the beauty of the cancellation! The terms simplify to:

We have transformed our complex product sequence into a sequence of reciprocals.

The Reciprocal Bridge

Now, let us connect this to the definition of Harmonic Progression (H.P.). A sequence is in H.P. if and only if the reciprocals of its terms are in A.P.

Since $a,b,c,d$ are in A.P., their reciprocals $\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}$ are in H.P. Our simplified sequence is $\frac{1}{d},\frac{1}{c},\frac{1}{b},\frac{1}{a}$, which is exactly the reverse of the sequence $\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}$.

The Final Twist

We know that if a sequence is in H.P., its reverse is also in H.P. Since $\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}$ is in H.P., then $\frac{1}{d},\frac{1}{c},\frac{1}{b},\frac{1}{a}$ is also in H.P.

This means our target sequence $abc,abd,acd,bcd$ must be in Harmonic Progression (H.P.).

It is a perfect, logical conclusion. We started with a complex product, used the scaling property of A.P., bridged the gap with the definition of H.P., and arrived at the answer with absolute certainty.

Keep this technique in your toolkit—whenever you see products of terms in a sequence, think about dividing by the product of all terms. It is a powerful way to reveal the hidden structure of the problem.