Question

If a, b, c are in A.P, then $$\dfrac{1}{{bc}}$$, $$\dfrac{1}{{ac}}$$, $$\dfrac{1}{{ab}}$$ will be in
(1) A.P
(2) G.P
(3) H.P
(4) None of these

Answer 1

Hint: Here in this question, given a one sequence of arithmetic progression (A.P) we need to find the type of another sequence. For this, we need to do any one of arithmetic operations like, Addition, subtraction, multiplication or division with the same number for all the numbers of A.P and on further simplification we get the required solution.

Complete step-by-step answer:
The general arithmetic progression is of the form $$a,a + d,a + 2d,...$$ where a is first term nth d is the common difference which is same between the distance on any two number in sequence otherwise the fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.
 The nth term of the arithmetic progression is defined as $${a_n} = a + (n - 1)d$$.
The common difference ‘$$d$$’ can be determined by subtracting the first term with the second term, second term with the third term, and so forth.
Now let us consider given question:
 If a, b, c are in A.P -----(1)
We know that, if we divide the numbers in AP with the same number, the resulting numbers will also be in AP.
Divide the given sequence i.e., (1) by abc.
$$ \Rightarrow $$ $$\dfrac{a}{{abc}}$$, $$\dfrac{b}{{abc}}$$, $$\dfrac{c}{{abc}}$$ are in A.P
On simplification, we get
$$\therefore $$ $$\dfrac{1}{{bc}}$$, $$\dfrac{1}{{ac}}$$, $$\dfrac{1}{{ab}}$$ are in A.P
Therefore, Option (1) is the correct answer.
So, the correct answer is “Option 1”.

Note: Remember, in an arithmetic sequence, the common difference (an addition or subtraction)
between any two consecutive terms of sequence is a constant or same. In any sequence of A.P if we divide or multiply all the numbers in A.P with the same number, the resulting numbers of sequence will also be in A.P.