Home
Class 12
MATHS
Prove that C0 – 2^2 C1 + 3² C2 – 4^2 C3 ...

Prove that `C_0 – 2^2 C_1 + 3² C_2 – 4^2 C_3 + ... +(-1)^n (n + 1)^2 C_n = 0` where `C_r = nC_r`

Text Solution

Verified by Experts

`C_(0) - 2^(2) C_(1) + 3^(2) . C_(2) - … + (-1)^(n) (n+1)^(2) . C_(n)`
`= sum_(r = 0 )^(n) *(-1)^(r) (r + 1)^(2)""^(n)C_(r) = sum_(r= 0) ^(n) (-1)^(r)(r^(2) + 2r + 1 ) ""^(n)C_(r)`
`= sum_(r = 0 )^(n) *(-1)^(r)r^(2). ""^(n)C_(r) +2 sum_(r= 0) ^(n) (-1)^(r)r. ""^(n)C_(r) + sum_(r=0)^(n) (-1)^(r). ""^(n)C_(r)`
`= sum_(r = 0 )^(n) *(-1)^(r)r(r - 1) ""^(n)C_(r) +3 sum_(r= 0) ^(n) (-1)^(r)r. ""^(n)C_(r) + sum_(r=0)^(n) (-1)^(r). ""^(n)C_(r)`
`= sum_(r = 0 )^(n) *(-1)^(r)n(n - 1) ""^(n - 2)C_(r-2) +3 sum_(r= 0) ^(n) (-1)^(r)n. ""^(n-1)C_(r-1) + sum_(r=0)^(n) (-1)^(r). ""^(n)C_(r)`
` = n(n- 1) {""^(n-2)C_(0) - ""^(n - 2)C_(0) - ""^(n-2)C_(1) + ""^(n-2)C_(2^(-))...+(-1)^(n) ""^(n-2)C_(n-2)}`
` = + 3n{-""^(n-1)C_(0)+ ""^(n - 1)C_(1) - ""^(n-1)C_(2)+ ...+(-1)^(n) ""^(n-1)C_(n-1)} + {""^(n)C_(0) -""^(n)C_(1) + ""^(n)C_(2) +...+ (-1)^(n)""^(n)C_(n)}
`= n(n-1). 0 + 3n .0 - AA n gt 2 = 0 , AA n gt 2 ` .
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    IIT JEE PREVIOUS YEAR|Exercise Topic 2 Properties of Binomial Coefficent Objective Questions I (Only one correct option) (Fill in the Blanks)|1 Videos

  • AREA

    IIT JEE PREVIOUS YEAR|Exercise AREA USING INTEGRATION|55 Videos

  • CIRCLE

    IIT JEE PREVIOUS YEAR|Exercise Topic 5 Integer Answer type Question|1 Videos

Similar Questions

Explore conceptually related problems

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

Prove that C_(0)-2^(2)C_(1)+3^(2)C_(2)-4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)xx C_(n)=0 where C_(r)=^(n)C_(r)

(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

Prove that "^nC_r + 2. ^nC_(r-1) + ^nC_(r-2) = ^(n+2)C_r

If (1+x)^n=C_0+C_1x+C_2x^2+…+C_nx^n show that C_1-2C_2+3C_3-4C_4+…+(-1)^(n-1) n.C_n=0 where C_r=^nC_r .

Prove that ^nC_0-^nC_1+^nC_2-^nC_3+....+(-1)^r ^C_r+....=(-1)^(r-1) ^(n-1)C_(r-1)

If n>2, then prove that C_(1)(a-1)-C_(2)xx(a-2)+...+(-1)^(n-1)C_(n)(a-n)=a, where C_(r)=^(n)C_(r)