JEE Main 2016 (1)

Algebra Level 5

If a , b , c a,b,c are three terms of an arithmetic progression such that a a is not equal to b b , then b c a b \frac{b-c}{a-b} is always

Take a , b , c a,b,c as real numbers.

Clarifications: a a , b b and c c may not be consecutive.

arithmetic progression


Try my set JEE Main 2016 .
Integer Natural number Irrational Rational

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2 solutions

Chew-Seong Cheong
Mar 21, 2016

If a a , b b and c c are three terms of an AP \text{AP} , then we can write:

{ b c = m d where m = { positive integer if b > c negative integer if b < c a b = n d where n = { positive integer if a > b negative integer if a < b and d = the common different > 0 \begin{cases} b - c = md \quad \text{where } m = \begin{cases} \text{positive integer if }b > c \\ \text{negative integer if }b < c \end{cases} \\ a - b = nd \quad \text{where } n = \begin{cases} \text{positive integer if }a > b \\ \text{negative integer if }a < b \end{cases} \end{cases} \text{ and } d = \text{the common different} > 0

Therefore b c a b = m d n d = m n \dfrac{b-c}{a-b} = \dfrac{md}{nd} = \dfrac{m}{n} a rational \boxed{\text{rational}} number.

Let us assume the three terms of A.P ( a , b , c a,b,c ) as ( x , x + d , x + 2 d x , x + d , x +2d ) where d 0 d \neq 0 .

b c a b = ( x + d ) ( x + 2 d ) x ( x + 2 d ) = d d = 1 \large \dfrac{b-c}{a-b} = \dfrac{(x + d) - (x+2d)}{x - (x + 2d)} = \dfrac{-d}{-d} = 1

1 is a rational,integer,complex, natural number.
What's wrong in my solution?

Akhil Bansal - 5 years, 2 months ago

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It's not given a,b,c are consecutive AP terms And complex number is a superset of rational number

Shivam Jadhav - 5 years, 2 months ago

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Please add the statement that a,b,c may not be consecutive terms. It becomes more clear to the readers.

Aditya Kumar - 5 years, 2 months ago

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@Aditya Kumar Yeah i agree with you

Prakhar Bindal - 5 years, 2 months ago

But a rational number must also be a complex number which is purely real.. Isn't it?

Rishabh Jain - 5 years, 2 months ago

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@Rishabh Jain Rational numbers are ratios of two integers (for example 1 3 , 123 1001 , 34 , . . . \frac{1}{3}, \frac{123}{1001}, 34,... ). Integers are rational numbers too. Complex numbers involve a real part and imaginary part. A complex number z z can be expressed as z = x + y i z = x+ yi where x x and y y are real, i = 1 i = \sqrt{-1} (the imaginary unit), and the real and imaginary parts are x x and y y respectively. Read more about it here .

Chew-Seong Cheong - 5 years, 2 months ago

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@Chew-Seong Cheong Note that the complex numbers include all the real numbers (with b = 0 b=0 in a + b i a+bi above)- That's clearly written on the page you mentioned... And rational numbers are real only.. I guess :-)

Rishabh Jain - 5 years, 2 months ago

@Chew-Seong Cheong Indeed we classify Complex number a + i b a+ib as purely real, purely imaginary or imaginary if b = 0 , a = 0 , b 0 b=0, a=0, b\neq 0 respectively. Real numbers fall under the first category..

Rishabh Jain - 5 years, 2 months ago

This information that a,b, and c are not consecutive must be given beforehand.

Akshay Yadav - 5 years, 2 months ago
Vineet PaHurKar
Apr 28, 2016

If a,b,and are in ap then , b-c=md where m is +ve integer a-b =nd where n is +ve integer so b-a/a-b=m/n m/n is rational...

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