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Prove that `(C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot`

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CENGAGE ENGLISH-BINOMIAL THEOREM-All Questions
  1. If k a n d n are positive integers and sk=1^k+2^k+3^k++n^k , then prov...

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  2. Prove that sum(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)^n Cr=1/n .

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  3. Prove that (C1)/1-(C2)/2+(C3)/3-(C4)/4++((-1)^(n-1))/n Cn=1+1/2+1/3++1...

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  4. Prove that sum(r=0)^n^n Crsinr xcos(n-r)x=2^(n-1)sin(n x)dot

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  5. Find the last two digits of the number (23)^(14)dot

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  6. Find the last two digits of the number 27^(27)dot

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  7. Find the number of nonzero terms in the expansion of (1+3sqrt(2)x)^9+(...

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  8. Find the value of (sqrt(2)+1)^6-(sqrt(2)-1)^6dot

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  9. Using binomial theorem (without using the formula for .^n Cr) , prove ...

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  10. Prove that .(r+1)*^n Cr-r*^n Cr+ ... +(-1)^r.^n Cr=(-1)^r.^(n-2)Crdot

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  11. Find the sum .^n C0+^n C4+^n C8 + . . .

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  12. Find the value of ^4nC0+^(4n)C4+^(4n)C8++""^(4n)C(4n) .

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  13. Find the coefficient of x^n in the polynomial (x+^n C0)(x+3^n C1)xx(x+...

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  14. If (1+x)^(15)=C0+C1x+C2x^2++C(15)x^(15), then find the value of C2+2C...

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  15. Prove that (.^n C0)/1+(.^n C2)/3+(.^n C4)/5+(.^n C6)/7+ . . . =(2^n)/(...

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  16. Find the sum sumsum(0lt=ilt=jlt=n)^n Ci^n Cj

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  17. Find the sum sumsum(i!=j)^n ^nCi^n Cj

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  18. Show that the integer next above (sqrt(3)+1)^(2m) contains 2^(m+1) as ...

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  19. Prove that (1^2)/3""^n C1+(1^2+2^2)/5^n C2(1^1+2^2+3^2)/7^n C3+ +(1^2+...

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  20. Prove that 1/(n+1)=(.^n C1)/2-(2(.^n C2))/3+(3(.^n C3))/4- . . . +(-1^...

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