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If n gt 2, then prove that C(1)(a-1) - ...

If `n gt 2`, then prove that `C_(1)(a-1) - C_(2) xx (a-2) + "….." + (-1)^(n-1) C_(n)(a-n) = a`, where `C_(r ) = .^(n)C_(r )`.

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

show that ^nC_r+ ^(n-1)C_(r-1)+ ^(n-1)C_(r-2)= ^(n+1)C_r

Show that .^(n)C_(r)+.^(n-1)C_(r-1)+.^(n-1)C_(r-2)=.^(n+1)C_(r) .

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2n.""^(4n-1)C_(2n-1) .

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

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CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
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  2. The coefficient of x^(n) in (1+x+(x^(2))/(2!) + (x^(3))/(3!) +"….." + ...

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  3. Evaluate the following .^3C2

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  4. Evaluate .^5C2

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  5. The value of sum(r=0)^(10)r""^(10)C(r)3^(r)(-2)^(10-r) is (a) 20 (b) 1...

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  6. Find n if n P1=2

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  7. Evaluate .^5P2

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  8. The value of (.^(n)C(0))/(n)+(.^(n)C(1))/(n+1)+(.^(n)C(2))/(n+2)+"..."...

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  9. The value of .^(20)C(0)+.^(20)C(1)+.^(20)C(2)+.^(20)C(3)+.^(20)C(4)+.^...

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  10. If (3+x^(2008)+x^(2009))^(2010)=a0+a1x^2+....+anx^n, a0-1/2 a1-1/2 a2+...

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  11. Find the seventh term of the G.P: 1, √3, 3, 3 √3, …..

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  12. Find the 10th term of the G.P. : 12, 4, 1 1/3,....

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  13. If (1+x)^(n) = C(0) +C(1)x+C(2)x^(2) + "….." + C(n)x^(n), then C(0)C(2...

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  14. The value of (lim)(n->oo)sum(r=1)^(n)(sum(t=0)^(r-1)1/(5^n).^n Cr.^r C...

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  15. If n gt 2, then prove that C(1)(a-1) - C(2) xx (a-2) + "….." + (-1)^(...

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  16. Find the nth term of the series: 1, 2, 4, 8, ……

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  17. The remainder, if 1+2+2^2+2^3+....+2^1999 in divided by 5 is

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  18. Largest real value for x such that underset(k=0)overset(4)sum((3^(4-k)...

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  19. If in the expansion of (a-2b)^(n), the sum of 5^(th) and 6^(th) terms ...

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  20. The number of real negative terms in the binomial expansion of (1+ix)^...

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