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

Show that `(2^(2) *C_(0) )/(1*2)+(2^(3)*C_(1))/(2*3)+(2^(4) *C_(2))/(3*4)+…+(2^(n+2)*C_(n))/((n+1)(n+2)) = (3^(n+2) - 2n-5)/((n+1)(n+2))`
Hence deduce that `(C_(0))/(1.2) -(C_(1))/(2.3) +(C_(2))/(3.4) -…=(1)/(n+2)`

A

`(3^(n+2)-2n-5)/((n+1)(n+2))`

B

`(3^(n+2)+2n+5)/((n+1)(n+2))`

C

`(3^(n+5) -5n +3)/((n+1)^2)`

D

none

Text Solution

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The correct Answer is:
A
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DIPTI PUBLICATION ( AP EAMET)-BINOMIAL THEOREM-EXERCISE 1B (BINOMIAL COEFFICIENTS)
  1. The sum of (n+1) terms of the series (C0)/(2)-(C1)/(3)+(C2)/(4)-(C3)/(...

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  2. (C0)/(2)+(C1)/(6)+(C2)/(12)+…..+ (Cn)/((n+1)(n+2))=

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

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  4. C0 C1+C1 C2 + C2 C3+…+ C(n-1) Cn=

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  5. If (C0 + C1) (C1+C2)…(C(n-1) + Cn)=k C0 C1 C2… Cn then k=

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  6. If (1)/(1!(n-1)!) + (1)/(3!(n-3)!) + (1)/(5!(n-5)!) +…. =

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  7. sum(r=0)^(n) (""^n Cr)^2 =

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  8. C0^2+3.C1^2+5. C2^2+….+(2n+1) .Cn^2=

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  9. If Cr denotes the binomial coefficient ""^n Cr then (-1) C0^2 + 2C1^2+...

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  10. If n is odd then C0^2 - C1^2+C2^2-…....+(-1)^n Cn^2=

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  11. If n is even then C0^2 - C1^2+C2^2-…....+(-1)^n Cn^2=

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  12. ""^((2n + 1))C0 - ""^((2n+ 1))C1 + ""^((2n + 1))C2 - ……+""^((2n + 1))C...

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

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  14. C1//1-C2//2+C3//3-C4//4+…+(-1)^(n-1) Cn//n=

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  15. C0 - 2^3 . C1 +3^2. C2 - …. + (-1)^n (n+1)^2 . (Cn)=

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

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  17. C0 - C2 + C4 - C6 +….....

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  18. C1 - C3 + C5 - C7 + …...

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  19. The term independent of x in (1+x)^n (1+(1)/(x))^n is

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  20. If a(n)=sum(r=0)^(n)(1)/(""^(n)C(r)) then sum(r=0)^(n)(r)/(""^(n)C(r))...

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