Question

Prove that:
[√(n/n+1)]>[(n+1)/(n+2)]
for all positive integer values of n.

Is the inequality valid for all positive rational values of n?

Answer 1

@ saruz, n is a positive integer

Answer 2

so, no question about n=-1 :)

Answer 3

that will be sqrt (1/2)

Answer 4

sqrt (1/2)>2/3

Answer 5

whats going on with me!

Answer 6

I think i should not answer any of the questions 2day, coz i m kinda out of my mind as my sat score is going to be released 2day....

Answer 7

yeah, it is valid for all positive integers...

Answer 8

well, we have to prove,
\[\sqrt{(n/n+1)}>(n+1)/(n+2)\]
square it in both sides and reverse it..
so, we have to prove
\[(n+1)/n<(n+2)^{2}/(n+1)^{2}\]
or,

Answer 9

\[1+1/n<1+2/(n+1)+1/(n+1)^{2}\]
simplifying, u can get, all u need to prove is,
\[n ^{?}+n-1>0 \forall n \in N\]
u can easily check that will be satisfied when

Answer 10

\[n>(\sqrt{5-1})/2\approx0.6\]

Answer 11

so, the inequation is valid for every positive integer

Answer 12

i got it i guess
look we need to prove n/(n+1) > (n+1)^2/(n+2)^2
here squaring is allowed as all numbers are going to be positive. and also we can cross multiply
n* (n+2)^2
---------- >1
(n + 1)^3

Answer 13

then if u subtract 1 from both sides, n simplify...
u get a quadratic in the numerator
and all nos. greater than the greater root will satisfy the eq.
i guess Arnab did the same thing in a shorter way