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Find the sum of the series C(1)+2^(2) *C...

Find the sum of the series `C_(1)+2^(2) *C_(2)x+3^(2)*C_(3)x^(2)+4^(2)*C_(4)x^(3)+….+n^(2)*C_(n) *x^(n-1)` and deduce `(n gt 2)` the value of `C_(1)-2^(2) *C_(2) +3^(2)-C_(3) - …+(-1)^(n-1) *n^(2)*C_(n)`

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The correct Answer is:
`n(1+x)^(n-2) (1+nx) and 0 `
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AAKASH SERIES-BINOMIAL THEOREM-EXERCISE - 1.2 (Level - 1)
  1. Prove the following: C(0)+2*C(1)+3*C(2)+…(n+1)*C(n)=(n+2).2^(n-1)

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  2. Prove the following: 2*C(0)+3*C(1)+4*C(2)+…+(n+2)*C(n)=(n+4).2^(n-1...

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  3. Prove that following i) 2.C(0)+5.C(1)+8.C(2)+…..+(3n+2)C(n)=(3n+4).2...

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  4. Prove the following: 3*C(1)+7*C(2)+11*C(3)+…( 4n-1)*C(n)=1+(2n-1)...

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  5. Prove the following: sum(r =0)^(n)(-1)^(r ) (3r+5).C(r )=0

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  6. Prove that following C(0)+(3)/(2).C(1)+(9)/(3).C(2)+(27)/(4).C(3)+……...

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  7. Prove the following: 1.2.C(1)+2.3C(2)+3.4.C(3)+….n(n+1).C(n) = n.(n+...

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  8. Prove that : Prove that (C(1))/(C(0))+2.(C(2))/(C(1))+3.(C(3))/(C(2)...

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  9. Prove the following: C(0)+(C(2))/(3) +(C(4))/(5) + … = (2^(n))/(n+1...

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  10. Prove that following (C(1))/(2)+(C(3))/(4)+(C(5))/(6)+(C(7))/(8)+……=...

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  11. Prove the following: C(0)-(C(1))/(2)+(C(2))/(3) - …+(-1)^(n)(C(n))/...

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  12. Prove that (""^(2n)C(0))^(2)-(""^(2n)C(1))^(2)+(""^(2n)C(2))-(""^(2n...

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  13. Prove that (C(0)+C(1))(C(1)+C(2))(C(2)+C(3))....(C(n-1)+C(n))=((n+1)...

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  14. Prove that (C(0) +C(1)+C(2)+….+C(n))^(2)=1 +""^(2n)C(1) +""^(2n)C(2)...

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  15. Use the identity (1+x)^(m)(1+x)^(n)=(1+x)^(m+n) to prove Vandermonde's...

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  16. Find the sum of the series C(1)+2^(2) *C(2)x+3^(2)*C(3)x^(2)+4^(2)*C(4...

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  17. Show that (2^(2) *C(0) )/(1*2)+(2^(3)*C(1))/(2*3)+(2^(4) *C(2))/(3*4)+...

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  18. Show that ""^(n)C(0) +""^((n+1))C(1) +""^((n+2))C(2) +…+""^((n+k))C(k)...

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  19. Show that ""^(30)C(0) +""^(30)C(1)+""^(30)C(2) +…+ ""^(30)C(14) =2^(29...

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