Question

If \[\frac{1}{{b - c}},\frac{1}{{c - a}},\frac{1}{{a - b}}\] be consecutive terms of an A.P., then \[{(b - c)^2},{(c - a)^2},{(a - b)^2}\] will be in
A. G.P.
B. A.P.
C. H.P.
D. None of these

Answer 1

Before we get started, let's talk about the AP, or arithmetic progression. An arithmetic progression, also known as an AP or arithmetic sequence, is a number sequence in which the difference between consecutive terms is constant.
Formula used:
\[a - d,a,a + d\]
Complete step-by-step solution:
We have been given in the question that,
The terms \[\frac{1}{{b - c}},\frac{1}{{c - a}},\frac{1}{{a - b}}\] are consecutive terms of an Arithmetic progression.
If \[{\rm{a}},{\rm{b}},{\rm{c}}\] is in Arithmetic progression, then we know that,
\[b - a = c - b = d\]
Where: \[d\]is the common difference
From the given information, it implies that
\[ \Rightarrow \frac{1}{{c - a}} - \frac{1}{{b - c}} = \frac{1}{{a - b}} - \frac{1}{{c - a}}\]
This means,
\[\left( {\frac{1}{{b - c}},\frac{1}{{c - a}},\frac{1}{{a - b}}} \right)\] are in AP.
The above expression can be written as,
\[ \Rightarrow \frac{{b - c - c + a}}{{(c - a)(b - c)}} = \frac{{c - a - a +b}}{{(a - b)(c - a)}}\]
Solve the above expression in fraction to simplify:
\[ \Rightarrow {a^2} - 2ac - {b^2} + 2bc = {b^2} - 2ab - {c^2} + 2ac\]
Cancel all the similar terms in the above equation, we obtain\[ \Rightarrow {a^2} - 2ac + {c^2} - {c^2} - {b^2} + 2bc = {b^2} - 2ab + {a^2} - {a^2} - {c^2} + 2ac\]
By square formula, restructure the above expression, we have\[ \Rightarrow \left( {{c^2} - 2ac + {a^2}} \right) - \left( {{b^2} - 2bc + {c^2}} \right) = \left( {{a^2} - 2ab + {b^2}} \right) - \left( {{c^2} - 2ac + {a^2}} \right)\]
Cancel the similar terms in the above equation and write using square formula:\[ \Rightarrow {(c - a)^2} - {(b - c)^2} = {(a - b)^2} - {(c - a)^2}\]
Now, the above expression becomes
\[{(b - c)^2},{(c - a)^2},{(a - b)^2}\]
Therefore, if\[\frac{1}{{b - c}},\frac{1}{{c - a}},\frac{1}{{a - b}}\]be consecutive terms of an A.P., then \[{(b - c)^2},{(c - a)^2},{(a - b)^2}\] will be in G.P.
Hence, the option A is correct.
Note:
To answer these types of questions, student must understand the arithmetic progression completely. In our daily lives, we encounter arithmetic progression quite frequently. One of the most important things to remember when answering these types of questions is that another formula can sometimes be used. So, don't get them mixed up. It also has a similar meaning to the one we used to answer the question. It's just that sometimes the question can be solved using this formula, so we use it.