Mathematical Induction

-終場ソ使者-

見到兩條數都幾有趣,睇下大家做唔做到
1)
For any positive integer n,let
U_n=1/2√3[(1+√3)ⁿ-(1-√3)ⁿ]
V_n=(1+√3)ⁿ+(1-√3)ⁿ
Prove by mathematical induction that U_n,V_n are integers and V_n is even.Hence, or otherwise, prove that U_2n is even.
2)(a)
Let f(x) be a convex function defined on [a,b], i.e.
f(x₁)+f(x₂)≤2f[(x₁+x₂)/2]
for any x₁,x₂∊[a,b].For each positive integer n, consider the statement
I(n): If x_i∊[a,b],i=1,2,...,n, then
f(x₁)+f(x₂)+...+f(x_n)≤nf[(x₁+x₂+...+x_n)/n]
(i) Prove by induction that I(2^k) is true for every positive integer k.
(ii) Prove that if I(n) (n≥2) is true, then I(n-1) is true.
(iii) Prove that I(n) is true for every positive n.
(b)
Prove that f(x)=sinx is convex on [0,π], and hence that
1/n(sinθ₁+sinθ₂+...+sinθ_n)≤sin[(θ₁+θ₂+...+θ_n)/n]  for 0≤θ_i≤π

另外我自己出左題(其實都唔算自己出)
Prove that √2 is irrational.

p445hkk20001

1/2√3[(1+√3)n-(1-√3)n] I not understand
can you explain more about?

-終場ソ使者-

OK,I can explain you further.
1÷(2√3)×[(1+√3)ⁿ-(1-√3)ⁿ]  (i.e. "/" means "÷")
Just like that.

-終場ソ使者-

其實要prove一樣野係xyz,
你可以assume佢唔係,
之後證明佢係contradiction,
呢個都係一個skill
呢題要運用mathematical logic

p445hkk20001

Un=1/2√3[(1+√3)n-(1-√3)n]
Vn=(1+√3)n+(1-√3)n
n=1 都好似balance唔到咁...

-終場ソ使者-

有balance的需要嗎?

p445hkk20001

我試下計一次
For convenience,we let x =1+√3 y=1-√3
when n=1
1+√3 +1-√3=2
when n=2
1+3+2√3+1-2√3+3=6
so (1+√3)^n+(1-√3)^n is a positive and even integer for n=1,2
assume for some positive integers k
[x^k+y^k]=N1 , [x^(k+1)+y^(k+1)]=N2
when N1 and N2 are positive and even integers
Then x^(k+2)+y^(K+2)
[x^(k+1)+y^(k+1)](x+y)-xy(x^k+y^k)
2[x^(k+1)+y^(k+1)]+2(x^k+y^k)=2(N1+N2)
so x^(k+2)+y^(k+2) is a positive and even integer since N1+ N2 is.
By indution ,(1+√3)^n+(1-√3)^n is a positive and even integer since for all positive and even integers n.

對不對?

wong87

加油努力~
看到英文便一頭無緒

p445hkk20001

回復 9# wong87


終於 有人理我-_-

仲有幾題 唔想做-_-