If `n=12 m(m in N),` prove that `^n C_0-(^n C_2)/((2+sqrt(3))^2)+(^n C_4)/((2+sqrt(3))^4)-(^n C_6)/((2+sqrt(3))^6)+ddot=((2sqrt(2))/(1+sqrt(3)))^ndot`

Prove that ^n C_1(^n C_2)(^n C_3)^3(^n C_n)^nlt=((2^n)/(n+1))^(n+1_C()_2),AAn in Ndot

Prove that (^n C_0)/1+(^n C_2)/3+(^n C_4)/5+(^n C_6)/7+.....+dot=(2^n)/(n+1)dot

Prove that ^n C_0 ^(2n)C_n- ^n C_1 ^(2n-2)C_n+ ^n C_2^(2n-4)C_n-=2^ndot

Prove that ^n C_0^n C_0-^(n+1)C_1^n C_1+^(n+2)C_2^n C_2-=(-1)^ndot

The value of log_((9)/(4))((1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3)))))...oo) is

Prove that n C_0+^n C_3+^n C_6+=1/3(2^n+2cos(npi)/3) .

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

The straight line 3x+4y-12=0 meets the coordinate axes at Aa n dB . An equilateral triangle A B C is constructed. The possible coordinates of vertex C (a) (2(1-(3sqrt(3))/4),3/2(1-4/(sqrt(3)))) (b) (-2(1+sqrt(3)),3/2(1-sqrt(3))) (c) (2(1+sqrt(3)),3/2(1+sqrt(3))) (d) (2(1+(3sqrt(3))/4),3/2(1+4/(sqrt(3))))

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

Prove that (C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot