Home
Class 12
MATHS
Prove that (.^n C0)/1+(.^n C2)/3+(.^n C4...

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

Text Solution

Verified by Experts

`S = (.^(n)C_(0))/(1)+(.^(n)C_(2))/(3)+(.^(n)C_(4))/(5)+(.^(n)C_(6))/(7)+"….."`
The general term of the series in
`(.^(n)C_(2r))/(2r+1) = (.^(n+1)C_(2r+1))/(n+1)`, where `r = 0,1,2,"….."`
`:. S = (1)/(n+1)[.^(n+1)C_(1)+.^(n+1)C_(3)+.^(n+1)C_(5)+"......."] = (2^(n))/(n+1)`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.7|9 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.8|10 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.5|8 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

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

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

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

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

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)+.......= (-1)^m((2sqrt(2))/(1+sqrt(3)))^ndot

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .

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

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.

Prove that (.^(2n)C_0)^2-(.^(2n)C_1)^2+(.^(2n)C_2)^2-..+(.^(2n)C_(2n))^2 = (-1)^n.^(2n)C_n .