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Prove that (2nC0)^2-(2nC1)^2+(2nC2)^2+.....

Prove that `(2nC_0)^2-(2nC_1)^2+(2nC_2)^2+.....+(2nC_2n)^2=(-1)^n2nC_n`

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Prove that sum_(r=0)^n r(n-r)(.^nC_ r)^2=n^2(.^(2n-2)C_n)dot

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PATHFINDER-BINOMIAL THEOREM-QUESTION BANK
  1. Prove that C1+C5+C9+....=1/2(2^(n-1)+2^(n//2)sin((npi)/4))

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  2. Prove that C1+C4+C7+....=1/3[2^n-2cos((n+1)/3pi)]

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  3. Prove that (2nC0)^2-(2nC1)^2+(2nC2)^2+.....+(2nC2n)^2=(-1)^n2nCn

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  4. The coefficient of x^9 in the expansion of (1+x)(1+x^2)(1+x^3)....(1+x...

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  5. The sum of coefficient of integral powers of x in the binomial expansi...

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  6. Number of irrational terms in the binomial expansion of (3^(1//5)+7^(1...

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  7. Coefficient of x^11 in the expansion of (1+x^2)^4(1+x^3)^7(1+x^4)^12 i...

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  8. Let S=2/1^nC0+2^2/2^nC1+2^3/3^nC2+....+2^(n+1)/(n+1)^nCn. Then S equal...

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  9. The coefficient of x^3 in the infinite series expansion of 2/((1-x)(2-...

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  10. The value of sum (nC1)^2+( nC2)^2+(nC3)^2+....+(nCn)^2 is

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  11. If the coefficient of x^8 in (ax^2+1/(bx))^13 is equal to the coeffici...

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  12. The coefficients of three consecutive terms of (1+x)^(n+5) are in the ...

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  13. The coefficient of the term independent of x in the expansion of [((x+...

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  14. Let n be a positive even integer. The ratio of the largest coefficient...

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  15. The sum of the series 1/(1x2)*(25)C0+1/(2x3)*25C1+1/(3x4)*25C2+....+1/...

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  16. If n is a possible integer, then (sqrt3+1)^(2n)-(sqrt3-1)^(2n) is

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  17. Let the coefficients of powers of x in the 2^(nd), 3^(rd) and 4^(th) t...

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  18. The sum of the series 1+(1/2) ^nC1 +(1/3) ^nC2 +....+(1/(n+1)) ^nCn ...

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  19. Let (1+x)^10=sum(r=0)^(10)crx^r and (1+x)^7=sum(r=0)^7drx^r . If P=sum...

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  20. The coefficient of x^10 in the expansion of 1+(1+x)+....+(1+x)^20 is

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