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Prove that ""^nC0+3""^nC1+5""^nC2+.........

Prove that `""^nC_0+3""^nC_1+5""^nC_2+........+(2n+1)""^nC_n= (n+1)2^n`.

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If (1 + x)^n= ""^nC_0+""^nC_1x+""^nC_2x^2+.............+ ""^nC_n x^n , prove that : ""^nC_1-2""^nC_2+3""^nC_3-.......+(-1)^(n-1) n ""^nC_n=0 .

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MODERN PUBLICATION-BINOMIAL THEOREM-EXERCISE
  1. Evaluate the following : ""^16C1+""^16C3+""^16C5+ ....... +""^16C15.

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  2. Prove that underset (r=0) overset (n) sum 3^r ""^nCr =4^n.

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  3. Prove that ""^nC0+3""^nC1+5""^nC2+........+(2n+1)""^nCn= (n+1)2^n.

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  4. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : C0-...

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  5. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : (1+...

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  6. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : C0-...

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  7. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : C0+...

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  8. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : 2C0...

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  9. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : 2C0...

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  10. If (1+x)^n =C0+C1 x+ C2 x^2 +....... Cnx^n prove the following : C0C...

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  11. Prove that C0Cr+C1 C(r+1)+ C2 C(r+2)+...............+c(n-r) Cn=((2n)!...

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  12. Find the coefficient of x^(n-r) in the expansion of (x+1)^n (1+x)^n. ...

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  13. Find the coefficient of x^4 in the expansion of (1+x)^n (1-x)^n. Deduc...

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  14. Write and simplify the term involving a^2b^5 in (a-2b)^4 (a+b)^3.

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  15. Expand using binomial theorem (1+x/2-2/x)^(4), x ne 0

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  16. Find an approximation of (0.99)^5 using the first three terms of its e...

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  17. If 6th, 7th, 8th and 9th terms of (x+y)^n are a, b,c and d respectivel...

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  18. Find the coefficient of x^10 in the expansion of : (1+x+x^2+x^3 +........

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  19. If kr is the coefficient of y^(r-1) in the expansion of (1+2y)^10, in ...

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  20. If x^p occurs in the expansion of (x^2+1/x)^(2n), prove that its coeff...

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