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Prove that ^n C0 .^n C0-^(n+1)C1 . ^n C1...

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

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Prove that ^mC_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-.....=(-1)^(m-1)n^mdot

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

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

Prove that .^n C_0+^n C_3+^n C_6+=1/3(2^n+2cos((npi)/3)) .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Prove that (.^n C_0)/1+(.^n C_2)/3+(.^n C_4)/5+(.^n C_6)/7+ . . . =(2^n)/(n+1)dot

Given, s_n=1+q+q^2++q^n ,S_n=1+(q+1)/2+((q+1)/2)^2++((q+1)/2)^n ,q!=1 prove that .^(n+1)C_1+^(n+1)C_2s_1+^(n+1)C_3s_2++^(n+1)C_(n+1)s_n=2^n S_ndot

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) *""^(2n)C_(n) - C_(1) *""^(2n-2)C_(n) + C_(2) *""^(2n-4) C_(n) -…= 2^(n)

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CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. Find the fourth term in the expansion of (1-2x)^(3//2)""dot

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

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  3. Prove that ^n C0 .^n C0-^(n+1)C1 . ^n C1+^(n+2)C2 . ^n C2-=(-1)^ndot

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  4. The sum of the coefficients of all the integral powers of x in the exp...

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  5. If the sum of the coefficient in the expansion of (alpha^2x^2-2alphax+...

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  6. Prove that sum(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10...

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  7. If (1+x-2x^2)^(20)=a0+a1x+a2x^2+a3x^3+...+a(40)x^(40), then find the v...

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  8. Find the sum of the series .^15 C0+^(15)C1+^(15)C2+...............+^(1...

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  9. Find the sum sum(k=0)^(10).^(20)Ck.

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  10. Find the sum of all the coefficients in the binomial expansion of (x^2...

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  11. If the sum of coefficient of first half terms in the expansion of (x+y...

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  12. Find the value of underset(p=1)overset(n)sum(underset(m=p)overset(n)su...

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  13. Show that the middle term in the expansion of (x+1)^(2n) is (1.3.5.......

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  14. If the middle term in the expansion of (x^2+1//x)^n is 924 x^6 , then...

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  15. The first three terms in the expansion of (1+a x)^n(n!=0) are 1,6x and...

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  16. If x^4 occurs in the rth term in the expansion of (x^4+1/(x^3))^(15), ...

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  17. Find the coefficient of x^(-10) in the expansion of (a/x+b x)^(12) .

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  18. Find the constant term in the expansion of (x-1/x)^6dot

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  19. If the coefficients of (r-5) th and(2r-1) th terms in the expansion of...

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  20. In ( 2^(1/3)+1/3^(1/ 3))^n if the ratio of 7th term from the beginnin...

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