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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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If n >2, then prove that C_1(a-1)-C_2xx(a-2)++(-1)^(n-1)C_n(a-n)=a ,w h e r eC_r=^n C_rdot

.^(n-1)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) +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) .

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

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. Find the sum .^1C0+^2C1+^3C2+....+^(n+1)Cn ,where Cr=^n Crdot

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  2. If (1+x+x^(2) + "……" + x^(n))^(n) = a(0) + a(1)x+a(2)x^(2) + "….." a(n...

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

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  4. Find the sum C0-C2+C4-C6+ ......... ,where Cr=^n Cr .

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  5. If x+y=1, prove that sum(r=0)^n .^nCr x^r y^(n-r) = 1.

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  6. Find the sum 3 C1 + 5C2

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  7. Prove that (.^(n)C(1))/(2) + (.^(n)C(3))/(4) + (.^(n)C(5))/(6) + "…." ...

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  8. If (1+x)^n=sum(r=0)^n^n Cr , show that C0+(C1)/2++(Cn)/(n+1)=(2^(n+1...

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  9. If underset(r=0)overset(2n)suma(r) (x-2)^(r) = underset(r=0)overset(2n...

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  10. 3^(2n+2)-8n-9 is divisible by

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  11. Statement 1: The number of distinct terms in (1+x+x^2+x^3+x^4)^(1000) ...

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  12. The product of 3rd and 8th term of a GP is 243. If its 4th term is 3. ...

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  13. The value of "^30C0.^30C10+^30C1.^30C11+^30C2.^30C(12)+……+^30C(20) . ^...

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  14. If f(x)=x^n ,f(1)+(f^1(1))/1+(f^2(1))/(2!)+........(f^n(1))/(n !),w h ...

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  15. The fractional part of =(2^(4n))/15 is

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  16. The value of .^(15)C(0)^(2)-.^(15)C(1)^(2)+.^(15)C(2)^(2)-"...."-.^(15...

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  17. If the sum of the coefficients in the expansion of (1-3x+10 x^2)^ni sa...

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  18. If (1+x-2x^2)^6=1+a1x+a2x^2+……+a(12)x^12 then

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  19. Maximum sum of coefficient in the expansion of (1-xsintheta+x^2)^n is

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  20. If the sum of the coefficients in the expansion of (a+b)^(n) is 4096, ...

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