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(C0)/(1) + (C2)/(3) + (C4)/(5) + ……+(C16...

`(C_0)/(1) + (C_2)/(3) + (C_4)/(5) + ……+(C_16)/(17) = `

A

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

B

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

C

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

D

`(1)/(n+1)`

Text Solution

Verified by Experts

The correct Answer is:
C
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Prove that (C_0)/(1)+ (C_2)/(3) + (C_4)/(5) + (C_6)/(7) +…….= (2^n)/(n+ 1)

Prove that (C_1)/(2) + (C_3)/(4) + (C_5)/(6) + (C_7)/(8) + …… = (2^n - 1)/(n+ 1)

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

C_0 + (C_1)/(2) (4) + (C_2)/(3) (16) + …………..+(C_n)/(n + 1) (2^(2n))

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

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)

DIPTI PUBLICATION ( AP EAMET)-BINOMIAL THEOREM-EXERCISE 1B (BINOMIAL COEFFICIENTS)
  1. Prove that : If n is a positive integer, then prove that C(0)+(C(1))...

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

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  3. (C0)/(1) + (C2)/(3) + (C4)/(5) + ……+(C16)/(17) =

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  4. C0 + (C1)/(2) + (C2)/(2^2) + (C3)/(2^3)+…..+(Cn)/(2^n)=

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  5. C3//4+C5//6+C7//8+….=

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  6. C0+(C1 x)/(2)+(C2 x^2)/(3)+…...+(Cn x^n)/(n+1)=

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  7. 2.C0 + (2^2)/(2).C1 + (2^3)/(3).C2 + ……+(2^11)/(11).C10 =

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

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  9. 2. C0+ 2^2 (C1)/(2)+2^3. (C2)/(3)+…....+2^(n+1). (Cn)/(n+1)=

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  10. (C0)/(2)+(C1)/(3)+(C2)/(4)+…...+(Cn)/(n+2)=

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  11. The sum of (n+1) terms of the series (C0)/(2)-(C1)/(3)+(C2)/(4)-(C3)/(...

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

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

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

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

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

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

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

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

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

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