Question

Given \[a + d > b + c\] where \[a,b,c,d\] are real numbers, then
A. \[a,b,c,d\] are in A.P.
B. \[\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}\] are in A.P.
C. \[(a + b),(b + c),(c + d),(a + d)\] are in A.P.
D. \[\frac{1}{{a + b}},\frac{1}{{b + c}},\frac{1}{{c + d}},\frac{1}{{a + d}}\] are in A.P.

Answer 1

Hint
Both rational and irrational numbers are considered to be real numbers. The integer 0 is a rational number and a real number, regardless of whether it is considered a natural number (as well as an algebraic number and a complex number). Since 0 is neither positive nor negative, it is typically shown in the middle of a number line.
Real numbers contain rational, irrational, and integer numbers, which is their primary distinction from the other types of numbers. The term "not real" or "non-real" refers to imaginary and unreal numbers. The number line cannot display non-real numbers.
Formula use:
If \[a,b,c\] are in AP
\[(b - a) = (c - b)\]
If \[a,b,c\] are in HP
\[\frac{1}{a} + \frac{1}{c} = \frac{2}{b}\]
Complete step-by-step solution
The given equation is \[a + d > b + c\]
This equation can be written as
\[ = > a + b + c + d > 2b + 2c\]
After taking two as a common factor, the equation is written as
\[ = > \frac{{a + c}}{2} + \frac{{b + d}}{2} > b + c\]
So, \[\frac{{a + c}}{2}\]\[ > b\] and \[\frac{{b + d}}{2}\]\[ > \]\[c\]
Hence, the points on series is \[A > H\]
b is determined as the Harmonic progression of a and c and the value of this series’ A, M is
\[\frac{{a + c}}{2}\]
c is determined as the Harmonic progression of b and d and the value of this series’ A, M is
\[\frac{{b + d}}{2}\]
So, the series of H.P has \[a,b,c,d\] and \[\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}\]are in the series of A.P.
Therefore, the correct option is B.
Note
It appears that the majority of other roots are equally illogical. In addition, the constants e and are irrational. Pi is an irrational number since it cannot be stated as a simple fraction. Every seemingly irrational number is real, as we are aware. Pi is thus a true number. Pi is an irrational number since it cannot be stated as a simple fraction. Every seemingly irrational number is real, as we are aware. Pi is thus a true number.