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Prove that 1-^n C1(1+x)/(1+n x)+^n C2(1+...

Prove that `1-^n C_1(1+x)/(1+n x)+^n C_2(1+2x)/((1+n x)^2)-^n C_3(1+3x)/((1+n x)^3)+....(n+1) terms =0`

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Prove that .^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2- .. =(-1)^n .

Show : x-^(n)C_(1)(x+y)+^(n)C_(2)(x+2y)-^(n)C_(3)(x+3y)+....=0

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1) . Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N .

Prove that (""^n C_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.

Prove that 1/(n+1)=(.^n C_1)/2-(2(.^n C_2))/3+(3(.^n C_3))/4- . . . +(-1^(n+1))(n*(.^n C_n))/(n+1) .

Find the value of (1+x)^(n)+^(n)C_(1)(1+x)^(n-1).(1-x)+^(n)C_(2)(1+x)^(n-2)(1-x)^(2)+.....+(1-x)^(n)

Prove that .^(n)C_(1) - (1+1/2) .^(n)C_(2) + (1+1/2+1/3) .^(n)C_(3) + "…" + (-1)^(n-1) (1+1/2+1/3 + "…." + 1/n) .^(n)C_(n) = 1/n

Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .

Prove that 1+(1+x)+(1+x+x^(2))+(1+x+x^(2)+x^(3))+...+ to n terms =(n)/(1-x)-(x(1-x^(n)))/(1-x)^(2)

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. If the third term in the expansion of (1+x)^mi s-1/8x^2, then find the...

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  2. Prove that sum(r=0)^n r(n-r)(.^nC r)^2=n^2(.^(2n-2)Cn)dot

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

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  4. Find the coefficient of x^(20) in (x^2+2+1/(x^2))^(-5)(1+x^2)^(40)dot

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  5. The number of terms in the expansion of (a+b+c)^n where ninN is

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  6. Find the coefficient of x^(50) in the expansion of (1+x)^(101)xx(1-x+x...

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  7. Find the coefficient of x^4 in the expansion of (2-x+3x^2)^6dot

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  8. Find the coefficient of x^k in1+(1+x)+(1+x)^2++(1+x)^n(0lt=klt=n)dot

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  9. The term independent of x in the expansion of (1+x+2x^3)(3/2(x^2)-1/(3...

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  10. If aa n db are distinct integers, prove that a-b is a factor of a^n-b^...

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  11. Find a, b and n in the expansion of (a+b)^(n) if the first three term...

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  12. Find the coefficient of x^(25) in expansion of expression sum(r=0)^(50...

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  13. If the sum of the coefficients of the first, second, and third terms ...

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

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  15. If every pair from among the equations x^2+a x+b c=0. x^2+b x+c a=0,a ...

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  16. Prove that ^mC1^n Cm-^m C2^(2n)Cm+^m C3^(3n)Cm-.....=(-1)^(m-1)n^mdot

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  17. Prove that ^nC0 "^(2n)Cn-^nC1 ^(2n-2)Cn +^nC2 ^(2n-4)Cn =2^n

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  18. Find the sum sum(r=0) .^(n+r)Cr .

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  19. Find the value of (sumsum)(0leiltjlen) (i+j)(""^(n)C(i)+""^(n)C(j)).

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  20. Find the sum sumsum(0lt=i<jlt=n-1)^n Cidot

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