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If (1+ x)^(n) = C(0) + C(1) x + C(2) x^(...

If `(1+ x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3)x^(3) + ...+ C_(n) x^(n) ,n` being even the value of
` C_(0) = (C_(0) + C_(1)) + (C_(0) + C_(1) + C_(2)) + ….+ (C_(0) + C_(1) + C_(2) + ….+ C_(n-1))` is equal to

A

`n 2^(n -1)`

B

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

C

` n 2 ^(n)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
a
We have,
`C_(0) + (C_(0)+C_(1))+…+(C_(0) +C_(1) +C_(2)+…+C_(0)+C_(1) +C_(2)+...+ C_(n-1))`
`nC_(0) +(n-1)C_(1) + (n-2) C_(2) +...+C_(n-1)`
`= sum_(r=1)^(n)(n-r)""^(n)C_(r)`
`= sum_(r=1)^(n)n. ""^(n)C_(r)- sum_(r=0)^(n) r. ""^(n)C_(r)`
`=n sum_(r=1)^(n) ""^(n)C_(r)-n sum_(r=0)^(n) ""^(n-1)C_(r-1)= n2^(n) - n2^(n-1) = n2^(n-1)` .
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OBJECTIVE RD SHARMA ENGLISH-BINOMIAL THEOREM AND ITS APPLCIATIONS -Section I - Solved Mcqs
  1. sum(r=0)^(n-1) (""^(n)C(r))/(""^(n)C(r) + ""^(n)C(r+1)) is equal to

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  2. If sum(i=1)^(n-1) ((""^(n)C(i-1))/(""^(n)C(i)+""^(n)C(i-1)))^(3) = (3...

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  3. If (1+ x)^(n) = C(0) + C(1) x + C(2) x^(2) + C(3)x^(3) + ...+ C(n) x^(...

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  4. The value of sum(r=1)^(10) r. (""^(n)C(r))/(""^(n)C(r-1) is equal to

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  5. 7^(103) when divided by 25 leaves the remainder .

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  6. The coefficient of x^(6) in the expansion of (1 + x + x^(2))^(-3) is

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  7. If (x^(2) + x)/(1-x) = a(1) x + a(2) x^(2) + ... to infty , |x| lt 1,...

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  8. The sum of the series ""^(4)C(0) + ""^(5)C(1) x + ""^(6)C(2) x^(2)...

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  9. the sum of the series ""^(2)C(0) + ""^(3)C(1) x^2 + ""^(4)C(2) x^(4)...

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  10. If Sn=sum(r=0)^n 1/(nCr) and tn=sum(r=0)^n r/(nCr), then tn/Sn=

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  11. If sn=sum(r < s) (1/(nCr)+1/(nCs)) and tn=sum(r < s)(r/(nCr)+s/(nCs)),...

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  12. The coefficient of x^5 in the expansion of (x^2-x-2)^5 is -83 b. -82 c...

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  13. If ar is the coefficient of x^r in the expansion of (1+x+x^2)^n, the...

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  18. the coefficient of x^(r) in the expansion of (1 - 4x )^(-1//2), is

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  19. In the expansion of (x^(2) + 1 + (1)/(x^(2)))^(n), n in N,

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