# If \[{p^{th}},{q^{th}},{r^{th}}\] and \[{s^{th}}\] terms of an A.P. be in G.P., then \[(p - q),(q - r),(r - s)\] will be in

A. G.P.

B. A.P.

C. H.P.

D. None of these

Answer

Verified

108.6k+ views

**Hint:**

We know that, \[{T_n} = a + \left( {n-1} \right)d\] is the nth term of an arithmetic progression (A.P) with first term 'a' and common difference- ’d’. The common difference can be positive, negative, or zero. And we can determine the nature of AP based on the value of the common difference d.

**Complete step-by-step solution:**

We have been given in the question, that

The \[{p^{th}},{q^{th}},{r^{th}}\] and \[{s^{th}}\] terms of an A.P. be in G.P

Let A be the first term D be the common difference of A.P.

We know that,

In terms of p-th term, we have

\[{a_p} = a + (p - 1)d\]

In terms of q-th term, we have

\[{a_q} = a + (q - 1)d\]

In terms of r-th term, we have

\[{a_r} = a + (r - 1)dn\]

In terms of s-th term, we have

\[{a_s} = a + (s - 1)dn\]

It is given that \[{a_p},{a_q},{a_r}\] and \[{a_s}\] is in Geometric progression.

Substitute all the values obtained from the above calculations, we obtain

Therefore, \[\frac{{{a_q}}}{{{a_p}}} = \frac{{{a_r}}}{{{a_q}}} = \frac{{{a_q} - {a_r}}}{{{a_p} - {a_q}}} = \frac{{q - r}}{{p - q}} \ldots \ldots \](1)

Now, we have to cancel the similar terms to simplify:

\[\frac{{{a_r}}}{{{a_q}}} = \frac{{{a_s}}}{{{a_r}}} = \frac{{{a_r} - {a_s}}}{{{a_q} - {a_r}}} = \frac{{r - s}}{{q - r}} \ldots \ldots \](2)

On simplifying, from equations (1) and (2), we get

\[\frac{{q - r}}{{p - q}} = \frac{{r - s}}{{q - r}}\]

Therefore, \[p - q,q - r\] and \[r - s\] are in Geometric progression.

**Hence, the option A is correct.**

**Note:**

The assumption that a common difference can never be negative or always positive is incorrect. So, student should determine whether the given terms have a mathematical formula to represent them; if so, it is a progression; otherwise, it is a sequence. However, a mathematical formula should be used to represent the given terms in progression. Identify the common difference between increasing and decreasing arithmetic progressions or determine whether a given sequence is increasing or decreasing.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

What is 1 divided by 0 class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How many crores make 10 million class 7 maths CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Draw a diagram of nephron and explain its structur class 11 biology CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

State the laws of reflection of light

Proton was discovered by A Thomson B Rutherford C Chadwick class 11 chemistry CBSE