We have to prove the inequality using A.M>G.M but I have no idea how to split the terms to do so
#Help with proving the above inequality
brave lake
craggy hemlockBOT
We have to prove the inequality using A.M>G.M but I have no idea how to split th...
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ionic crownBOT
wispy flicker
Bruh
Well if I remember correctly, it’s that …
Ok for monotonic increasing sequences $a_n, b_n$ we have that $$\sum_{k=1}a_kb_k \geq \sum_{k=1}a_kb_{\pi(k)}$$ for any permutation $\pi$ on ${1,…,n}$
ionic crownBOT
ℝανι-sαη
wispy flicker
You can assume WLOG that b>a
brave lake
Wait so it can't be done through the sequence and series formulae?
brave lake
Permutation ineq might play a role here
I haven't done permutation and combination yet
drifting solstice
See if you can get some Information from the fact that a and b are natural numbers and a ≠b
brave lake
See if you can get some Information from the fact that a and b are natural numbe...
Yeah but how will you separate them?
I guess you have to separate a b times and b a times
drifting solstice
Yeah but how will you separate them?
Is it necessary to seperate the terms?
I think not
brave lake
Is it necessary to seperate the terms?
Yeah if ur doing it through am>gm no?
To get a power b terms u have to separate it b times
Like for example if it was a²b² we would write a/2 + a/2 + b/2 + b/2
Then do the am >gm method
drifting solstice
Try to do some algebraic manipulations from AM>GM
brave lake
I mean we can write it as a/b + a/b ... b times
But I'm not getting anything from that method
Unless u need to shift the powers or smth
drifting solstice
Unless u need to shift the powers or smth
Well, you can raise terms to some power which preserves the inequality
brave lake
How though
drifting solstice
Full solutions aren't allowed, sadly, you can only get hints
You have to do the work yourself
drifting solstice
No but tell me a method
- Raise terms to appropriate power
- Do manipulations using the fact that a, b ∈ℕ and a ≠ b
- See how you can obtain the terms
brave lake
I tried algebraic manipulation but I got no results
brave lake
1. Raise terms to appropriate power
2. Do manipulations using the fact that a, b...
Can u give an example of the 1st
Like squaring both sides of some sorts
?
If we want to get the terms we have to split a b times and b a times
drifting solstice
Can u give an example of the 1st
Such as, 1/ab?
drifting solstice
Try ab also
brave lake
Wait ur using the am>gm relation right?
brave lake
Try ab also
Alr
drifting solstice
Wait ur using the am>gm relation right?
Yea
How one can prove it without AM>GM
ebon lavaBOT
@brave lake
brave lake
+close