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Prove that "^(2n)C0 + ^(2n)C2 + .... + ^...

Prove that `"^(2n)C_0 + ^(2n)C_2 + .... + ^(2n)C_(2n) = 2^(2n-1)`

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`"^(2n)C_0 + ^(2n)C_2 + .... + ^(2n)C_(2n)` = 2^(2n-1)`
We know that `(1+x)^(2n) = "^(2n)C_0 + ^(2n)C_1 x + ^(2n)C_2 x^2 + ... ^(2n)C_(2n) x^n` ... (1)
Putting x = 1
we get putting x = 1 we get
`("^(2n)C_0 + ^(2n)C_2 + ^(2n)C_4 + .... + ^(2n)C_(2n)) - (^(2n)C_1 + ^(2n)C_3 + ... + ^(2n)C_(2n-1))` = 0
therefore ^(2n)C_0 + ^(2n)C_2 + .... + ^(2n)C_(2n)` = `"^(2n)C_1 + ^(2n)C_3 + .... + ^(2n)C_(2n-1)`
= `2^(2n)/2` = `2^(2n-1)`
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MBD PUBLICATION-Elements of Mathematics-QUESTION BANK
  1. An expression of the form (a+b+c+d+ .... ) consisting of sum of many d...

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  2. State and prove a multinomial Theorem.

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  3. Prove that "^(2n)C0 + ^(2n)C2 + .... + ^(2n)C(2n) = 2^(2n-1)

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  4. Prove that "^(2n)C1 + ^(2n)C3 + .... + ^(2n)C(2n-1) = 2^(2n-1)

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  5. Find the sum of C1 + 2C2 + 3C3 + .... + nCn

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  6. Find the sum of C0 + 2C1 + 3C2 + .... + (n+1)Cn

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  7. Compute ((1+k)(1+k/2) ..... (1+k/n))/((1+n)(1+n/2) ..... (1+n/k))

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  8. Show that C0 C1 + C1 C2 + C2 C3 + .... + C(n-1) Cn = (2n!)/((n-1)!(n...

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  9. C0 C1 + C1 C2 + .... + C(n-1) Cn = (2^n.n.1.3.5... (2n-1))/(n+1)

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  10. Show that 3C0-8C1 + 13C2 - 18C3 + ..... + (n+1)^(th) term = 0

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  11. Show that C0 n^2 + C1 (2-n)^2 + C2 (4-n)^2 + .... + Cn (2n-n)^2 = n.2^...

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  12. Show that C0 + 3C1 + 5C2 + .... +(2n+1) Cn = (n+1)(2^n)

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  13. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

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  14. Find the sum of the following 1.2 C(2) + 2.3 C(3) + ... + (n-1)nCn

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  15. Find the sum of the following C1 + 2^2 C2 + 3^2 C3 + ... + n^2Cn

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  16. Find the sum of the following C1 - 2C2 + 3C3 - ..... + n(-1)^(n-1) Cn

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  17. Show that C1^2 + 2C2^2 + 3C3^2 + ... + "^nCn^2 = ((2n-1!))/{(n-1)!}^2

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  18. Show that C2 + 2C3 + 3C4 + ... + (n-1)Cn = 1+ (n-2) 2^(n-1)

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  19. prove that :-C1 - 1/2 C2 + 1/3 C3 + ... + (-1)^(n+1) 1/n Cn = 1+1/2 + ...

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  20. C0 C1 + C1 C2 + .... + C(n-1) Cn = (2^n.n.1.3.5... (2n-1))/(n+1)

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