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

Show that `C_0 n^2 + C_1 (2-n)^2 + C_2 (4-n)^2 + .... + C_n (2n-n)^2` = `n.2^n`

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`C_0 n^2 + C_1 (2-n)^2 + C_2 (4-n)^2 + .... + C_n (2n-n)^2`
`n^2(C_0 + C_1 + .... + C_n) + (2^2C_1 + 4^2C_2 + ...... + (2n)^2C_n) - 4n(C_1 + 2C_2 + 3C_3 + ... + nC_n)`
= `n^2.2^n + 4n(n+1)2^(n-2) - 4n.n.2^(n-1`)
`2^(n-1)(2n^2 + 2n^2 + 2n - 4n^2)
= `2n.2^(n-1)` = `n2^n`
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MBD PUBLICATION-Elements of Mathematics-QUESTION BANK
  1. Prove that "^(2n)C1 + ^(2n)C3 + .... + ^(2n)C(2n-1) = 2^(2n-1)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  18. The sum 1/(1!9!) + 1/(3!7!) + ... + 1/(7!3!) + 1/(9!1!) can be written...

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  19. Using binomial theorem show that 1^(99) + 2^(99) +3^(99) + 4^(99) + 5^...

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  20. Using the binomial theorem show that 1^(99) + 2^(99) +3^(99) + 4^(99) ...

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