Mathematical Induction
見到兩條數都幾有趣,睇下大家做唔做到1)
For any positive integer n,let
Un=1/2√3[(1+√3)n-(1-√3)n]
Vn=(1+√3)n+(1-√3)n
Prove by mathematical induction that Un,Vn are integers and Vn is even.Hence, or otherwise, prove that U2n is even.
2)(a)
Let f(x) be a convex function defined on , i.e.
f(x1)+f(x2)≤2f[(x1+x2)/2]
for any x1,x2∊.For each positive integer n, consider the statement
I(n): If xi∊,i=1,2,...,n, then
f(x1)+f(x2)+...+f(xn)≤nf[(x1+x2+...+xn)/n]
(i) Prove by induction that I(2k) 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 , and hence that
1/n(sinθ1+sinθ2+...+sinθn)≤sin[(θ1+θ2+...+θn)/n]for 0≤θi≤π
另外我自己出左題(其實都唔算自己出)
Prove that √2 is irrational. 1/2√3[(1+√3)n-(1-√3)n] I not understand
can you explain more about? OK,I can explain you further.
1÷(2√3)×[(1+√3)n-(1-√3)n](i.e. "/" means "÷")
Just like that. 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 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... 有balance的需要嗎? 本帖最後由 p445hkk20001 於 17-4-2010 22:55 編輯
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
=N1 , =N2
when N1 and N2 are positive and even integers
Then x^(k+2)+y^(K+2)
(x+y)-xy(x^k+y^k)
2+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.
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