Calculating the Ratio \(\frac{{a - b}}{{b - c}}\) for AP, GP, and HP

This question asks us to determine the possible values of the ratio \(\frac{{a - b}}{{b - c}}\) given that three numbers a, b, and c are in either an Arithmetic Progression (AP), a Geometric Progression (GP), or a Harmonic Progression (HP).

We will analyze each case separately based on the definitions of AP, GP, and HP.

Case 1: a, b, c are in Arithmetic Progression (AP)

If a, b, and c are in AP, the difference between consecutive terms is constant. This means:

\[ b - a = c - b \]

Rearranging this equation, we can write:

\[ a - b = - (b - a) \]

and

\[ b - c = - (c - b) \]

From the property \(b - a = c - b\), if we move terms, we get \(a - b = b - c\).

The ratio we need to calculate is \(\frac{{a - b}}{{b - c}}\). Substituting \(a - b = b - c\) into the ratio (assuming \(b - c \neq 0\), which implies a, b, c are not all equal):

\[ \frac{{a - b}}{{b - c}} = \frac{{b - c}}{{b - c}} = 1 \]

So, when a, b, c are in AP (and are not all equal), the value of \(\frac{{a - b}}{{b - c}}\) is 1.

Case 2: a, b, c are in Geometric Progression (GP)

If a, b, and c are in GP, the ratio of consecutive terms is constant. This means:

\[ \frac{b}{a} = \frac{c}{b} \]

This property implies \(b^2 = ac\).

We want to evaluate \(\frac{{a - b}}{{b - c}}\). Let's manipulate this expression:

\[ \frac{{a - b}}{{b - c}} = \frac{a \left( 1 - \frac{b}{a} \right)}{b \left( 1 - \frac{c}{b} \right)} \]

Since \(\frac{b}{a} = \frac{c}{b}\), let's call this common ratio 'r'. So \(b = ar\) and \(c = br = ar^2\). Substituting these into the ratio:

\[ \frac{{a - b}}{{b - c}} = \frac{{a - ar}}{{ar - ar^2}} = \frac{{a(1 - r)}}{{ar(1 - r)}} \]

Assuming \(r \neq 1\) (which means a, b, c are not all equal), we can cancel the term \((1 - r)\):

\[ \frac{{a(1 - r)}}{{ar(1 - r)}} = \frac{a}{ar} = \frac{1}{r} \]

Now, let's express \(\frac{1}{r}\) in terms of a and b. Since \(r = \frac{b}{a}\), then \(\frac{1}{r} = \frac{a}{b}\).

Alternatively, using the first form \(\frac{{a(1 - b/a)}}{{b(1 - c/b)}}\), since \(\frac{b}{a} = \frac{c}{b}\), assuming \(\frac{b}{a} \neq 1\) (i.e., a ≠ b), we get:

\[ \frac{{a \left( 1 - \frac{b}{a} \right)}}{{b \left( 1 - \frac{b}{a} \right)}} = \frac{a}{b} \]

So, when a, b, c are in GP (and are not all equal), the value of \(\frac{{a - b}}{{b - c}}\) is \(\frac{a}{b}\).

Case 3: a, b, c are in Harmonic Progression (HP)

If a, b, and c are in HP, their reciprocals, \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\), are in AP. This means:

\[ \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} \]

Combining terms, we get:

\[ \frac{{a - b}}{{ab}} = \frac{{b - c}}{{bc}} \]

We want to find the ratio \(\frac{{a - b}}{{b - c}}\). From the HP property equation above, we can rearrange to isolate this ratio (assuming \(b - c \neq 0\) and \(ab \neq 0\), \(bc \neq 0\), which is true if a, b, c are distinct and non-zero, required for HP):

\[ \frac{{a - b}}{{b - c}} = \frac{{ab}}{{bc}} \]

Assuming \(b \neq 0\), we can cancel 'b' from the numerator and denominator on the right side:

\[ \frac{{a - b}}{{b - c}} = \frac{a}{c} \]

So, when a, b, c are in HP (and are distinct and non-zero), the value of \(\frac{{a - b}}{{b - c}}\) is \(\frac{a}{c}\).

Summary of Results for \(\frac{{a - b}}{{b - c}}\)

Based on our calculations for each type of progression:

Thus, if a, b, and c are in AP or GP or HP, the possible values for \(\frac{{a - b}}{{b - c}}\) are \(1\), \(\frac{a}{b}\), or \(\frac{a}{c}\).

Revision Table: AP, GP, HP Properties

Additional Information on Progressions

Progressions describe sequences where terms follow a specific pattern. Understanding the relationships between consecutive terms is key to solving problems involving AP, GP, and HP.

• Arithmetic Progression (AP): Each term after the first is obtained by adding a fixed constant, called the common difference (d), to the preceding term. General form: \(a, a+d, a+2d, \dots\).
• Geometric Progression (GP): Each term after the first is obtained by multiplying the preceding term by a fixed non-zero constant, called the common ratio (r). General form: \(a, ar, ar^2, \dots\).
• Harmonic Progression (HP): A sequence is in HP if the reciprocals of its terms are in AP. There is no general formula for the n-th term of an HP.

The question highlights how the defining properties of these progressions directly influence algebraic relationships between their terms.