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If (1+x)^n=sum(r=0)^nCrx^r prove that (2...

If `(1+x)^n=sum_(r=0)^nC_rx^r` prove that `(2^2C_0)/(1 . 2)+(2^3C_1)/(2 . 3)+....+(2^(n+2) C_n)/((n+1)(n+2))=(3^(n+2)-2n-5)/((n+1)(n+2))`

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PATHFINDER-BINOMIAL THEOREM-QUESTION BANK
  1. Find the coefficient of x^4 in the expansion of (1+x-2x^2)^7

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  2. If the coefficients of four consecutive terms in the expansion of (1+x...

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  3. If (1+x)^n=sum(r=0)^nCrx^r prove that (2^2C0)/(1 . 2)+(2^3C1)/(2 . 3)+...

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  4. If p+q=1, then show that sum(r=0)^nr^2^nCrp^rq^(n-r)=npq+n^2p^2

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  5. Prove that C1+C5+C9+....=1/2(2^(n-1)+2^(n//2)sin((npi)/4))

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

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

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

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

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

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  11. 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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  12. 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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  13. The coefficient of x^3 in the infinite series expansion of 2/((1-x)(2-...

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

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

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

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

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

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

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

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