If r is a rational number and n a positive integer, then which of the following statement must be true for t, if t = {(r^n) - 1}/ (r - 1)?
I. t is rational.
II. t is positive.
III. t is negative.
A. I only
B. II only
C. III only
D. I & II only
E. I & III only
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r is a rational
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- sanju09
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Sanjeev K Saxena
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Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Hey,
1. If 0<r<1 then;
r-1 is negative
r^n1 is also negative as for 0<r<1, r^n<r
This means t is positive
2. If r<-1
r-1 will be negative
r^n-1 can be positive or negative depending on n (even or odd)
This means t can either be negative or positive
Thus conclusively nothing can be said about t's sign it can be either negative or positive
Now as r is a rational number it can be written as p/q
Substituting in the given equation
t= (p^n-q^n)/(p-q)*q^n-1
This can be expressed finally as p'/q'
Hence, t is rational
Answer: A -I only
1. If 0<r<1 then;
r-1 is negative
r^n1 is also negative as for 0<r<1, r^n<r
This means t is positive
2. If r<-1
r-1 will be negative
r^n-1 can be positive or negative depending on n (even or odd)
This means t can either be negative or positive
Thus conclusively nothing can be said about t's sign it can be either negative or positive
Now as r is a rational number it can be written as p/q
Substituting in the given equation
t= (p^n-q^n)/(p-q)*q^n-1
This can be expressed finally as p'/q'
Hence, t is rational
Answer: A -I only
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we know, that (a^n-b^n)is always divisible by (a-b) => the result of division will always be rationalt = {(r^n) - 1}/ (r - 1)
however, r being -ve or =ve depends on n being odd or even.
hence (A) I only