By considering the roots of the equation tan5θ=tan5β (where β is a constant), or otherwise, show that
4
Σcot(β+kπ/5) = 5cot5
k=0
2. Sequence {qn}, n=1,2,3,… satisfies the following conditions:
0< qn <1 and (1- qn) qn+1 >1/4.
a)Prove that qn+1 > qn, n=1,2,3,…
b)Show that lim n→∞(qn) exists and find the limit
3. a) Let a,b,c be real numbers.
i) Show that a^2+b^2+c^2≧ab+bc+ca.
ii) Hence deduce that if a+b+c>0, then a^3+b^3+c^3≧3abc.
b) Let∣x∣< ln2.
i) Show that (e^-x)^1/3 + (2 - e^x)^1/3 + (e^x - e^-x + 1)≦1.
Help!
1. Show that tan5θ = (5cot^4θ-10cot^2θ+1) / (cot^5θ-10cot^3θ+5cotθ) By considering the roots of the equation tan5θ=tan5α (where α is a constant), or otherwise, show that summation{ cot(α+kπ/5) }(where k = 0 to 4) = 5cot5α
2. Sequence {qn}, n=1,2,3, ... satisfies the following conditions: 0< qn <1 and (1- qn) qn+1 >1/4. a)Prove that qn+1 > qn, n=1,2,3, ... b)Show that lim n→∞(qn) exists and find the limit.
3. a) Let a,b,c be real numbers. i) Show that a^2+b^2+c^2≧ab+bc+ca. ii) Hence deduce that if a+b+c>0, then a^3+b^3+c^3≧3abc. b) Let |x|< ln2. i) Show that (e^-x)^1/3 + (2 - e^x)^1/3 + (e^x - e^-x + 1)≤1.
3. a) Let a,b,c be real numbers. i) Show that a^2+b^2+c^2=>ab+bc+ca. ii) Hence deduce that if a+b+c>0, then a^3+b^3+c^3=>3abc. b) Let |x|< ln2. i) Show that (e^-x)^1/3 + (2 - e^x)^1/3 + (e^x - e^-x + 1)≤1.
2. Sequence T(n), n=1,2,3, ... satisfies the following conditions: 0< T(n) <1 and [1- T(n)][T(n+1)] >1/4. a)Prove that T(n+1) > T(n), n=1,2,3, ... b)Show that lim n→∞T(n) exists and find the limit. I also want to know how the roots in question 1 are found. Thx~
