`(C_0)/1+(C_2)/3+(C_4)/5+(C_6)/7+.....=`

If C_(0)+C_(1)+C_(2)+......+C_(n)=128 then C_(0)-(C_(1))/(2)+(C_(2))/(3)-(C_(3))/(4)+.....=

If C_(0)+C_(1)+C_(2)+......+C_(n)=128 then C_(0)-(C_(1))/(2)+(C_(2))/(3)-(C_(3))/(4)+......=

(8_(C_(0)))/(1)+(8_(C_(2)))/(3)+8_(C_(4))/5+(8_(C_(6)))/(7)+(8_(C_(8)))/(9) =

(C_(0)-C_(2)+C_(4)-C_(6)......)^(2)+(C_(1)-C_(3)+C_(5)-C_(7).........)^(2)=2^(n)

If C_(0),C_(1),C_(2)...,C_(n), denote the binomial coefficients in the expansion of (1+x)^(n), then (C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+...... is equal to

Prove that (.^(n)C_(0))/(1)+(.^(n)C_(2))/(3)+(.^(n)C_(4))/(5)+(.^(n)C_(6))/(7)+"....."+= (2^(n))/(n+1)

If for z as real or complex . (1+z^(2) + z^(4))^(8) = C_(0) C_(1) z^(2) C_(2) z^(4) + …+ C_(16) z^(32) , prove that C_(0) + C_(3) + C_(6) + C_(9) + C_(12) + C_(15) + (C_(2) + C_(5) + C_(8) + C_(11) + C_(14)) + (C_(1) + C_(4) + C_(7) + C_(10) + C_(16)) omega^(2) = 0 , where omega is a cube root of unity .

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + C_(3) x^(3) + C_(4) x^(4) + ..., find the values of (i) C_(0) - C_(2) + C_(4) - C_(0) + … (ii) C_(1) - C_(3) + C_(5) - C_(7) + … (iii) C_(0) + C_(3) + C_(6) + …

" 0.Show that "C_(0)-4.C_(1)+7.C_(2)-10.C_(3)+.....=0