Question

If \[a,b,c\] be in G.P. and \[a + x,b + x,c + x\] in H.P., then the value of \[x\] (is \[a,b,c\] are distinct numbers)
A. c
B. b
C. a
D. None of these

Answer 1

Hint
A number in a set that is not equal to another number is referred to as a distinct number in mathematics. If a mathematical equation has real roots, then its solutions or roots are real numbers. If an equation has distinct roots, we contend that all of its solutions or roots are not equal. A quadratic equation has real and distinct roots if its discriminant is bigger than.
A number in a set that is not equal to another number is referred to as a distinct number in mathematics.
Formula used:
 \[a,b,c\] be in G.P.
 \[{b^2} = ac\]
Apply the relationship of H.P on \[a,b ,c \]
\[\frac{1}{{a }},\frac{1}{{b }}\]and \[\frac{1}{{c }}\] are in A.P

Complete step-by-step solution
The given relationship on G.P. is \[a,b,c\]
\[a + x,b + x,c + x\] in H.P.
Here, find the value of x
As \[a,b,c\] be in G.P.
So, \[{b^2} = ac\] ---(1)
Apply the relationship of H.P on \[a + x,b + x,c + x\]
\[\frac{1}{{a + x}},\frac{1}{{b + x}}\]and \[\frac{1}{{c + x}}\] are in A.P
So, \[\frac{1}{{b + x}} - \frac{1}{{a + x}} = \frac{1}{{c + x}} - \frac{1}{{b + x}}\]
It can also be written as
\[\frac{{(a + x) - (b + x)}}{{(a + x)(b + x)}} = \frac{{(b + x) - (c + x)}}{{(c + x)(b + x)}}\]
Simplify the equation
\[ = > \frac{{(a - b)}}{{(a + x)}} = \frac{{(b - c)}}{{(c + x)}}\]
This is by cancelling \[(b + x)\] from both the sides
\[ = > (a - b)(c + x) = (a + x)(b - c)\]
\[ = > ac + ax - bc - bx = ab - ac + bx-cx\]
The equation is simplified as
\[ = > 2ac - bc - ab + ax - 2bx + cx = 0\]
\[ = > 2{b^2} - bc - ab + (a - 2b + c)x = 0\]
Which is similar to the formula \[{b^2} = ac\]
\[ = > b(2b - c - a) = (2b - a - c)x\]
Cancel out \[(2b - c - a)\] on both the sides
\[ = > b = x\]
Therefore, the correct option is B.
Note
In an arithmetic progression (AP), the differences between every two consecutive terms are all the same, whereas in a geometric progression, the ratios of every two successive terms are all the same (GP). The common ratio is a constant value. An infinitely long geometric progression is what is meant by the term. In other words, the last term is not finite. The amount by which the values in a series are regularly multiplied is known as the common ratio.