Find the sum `2C_0+(2^3)/2C_1+(2^3)/3C_2+(2^4)/4C_3++(2^(11))/(11)C_(10)dot`

Find the sum 2..^(10)C_(0) + (2^(2))/(2).^(10)C_(1) + (2^(3))/(3).^(10)C_(2)+(2^(4))/(4).^(10)C_(3)+"...."+(2^(11))/(11).^(10)C_(10) .

Find the sum C_0+3C_1+3^2C_2++3^n C_ndot

Find the sum 3^n C_0-8^n C_1+13^n C_2xx^n C_3+dot

Find C_0 + (C_1)/(2) + (C_2)/(2^2) + (C_3)/(2^3)+…..+(C_n)/(2^n)= ?

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + … + C_(n) x^(n) , Show that (2^(2))/(1*2) C_(0) + (2^(3))/(2*3) C_(1) + (2^(4))/(3*4)C_(2) + ...+ (2^(n+2)C_n)/((n+1)(n+2))= (3^(n+2)-2n-5)/((n+1)(n+2))

If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n in N prove that (a) 3 C_0- 8C_1+13C_2-18C_3+...."upto" (n+1) term=0 if n ge 2 (b ) 2C_0+2^2(C_1)/(2)+2^3(C_2)/(3)+2^4C_(3)/4+....+2^(n+1)(C_n)/(n+1)=(3^n+1-1)/(n+1) ( c) C_0^2+(C_1^2)/2+C_2^2/3+....+C_n^2/(n+1)=((2n+1)!)/(((n+1)!)^2)

the sum of the series ""^(2)C_(0) + ""^(3)C_(1) x^2 + ""^(4)C_(2) x^(4)+""^(5)C_(3)x^(6) +... to infty , is

Find the value of .^20 C_0-(.^20C_1)/2+(.^20C_2)/3-(.^20C_3)/4+...

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

Find the sum of the series ^10 C_0+^(10)C_1+^(10)C_2++^(10)C_7dot