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Prove that (C0)/(1)+ (C2)/(3) + (C4)/(5)...

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

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Prove that (C_1)/(2) + (C_3)/(4) + (C_5)/(6) + (C_7)/(8) + …… = (2^n - 1)/(n+ 1)

C_0 + (C_1)/(2) (4) + (C_2)/(3) (16) + …………..+(C_n)/(n + 1) (2^(2n))

Prove the following: C_(0)+(C_(2))/(3) +(C_(4))/(5) + … = (2^(n))/(n+1)

Prove that following (C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+(C_(7))/(8)+……=(2^(n)-1)/(n+1)

Prove that (C_0 + C_1) (C_1 + C_2) …..(C_(n-1) + C_n) = ((n+1)^n)/(n!) (C_1.C_2.C_3……C_n)

Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).

With usual notations prove that C_1/C_0 + 2. C_2/C_1 + 3.C_3/C_2 + ……+n.(C_n)/(C_(n-1)) = (n(n +1))/(2) Hence prove that (15C_1)/(15C_0) + 2.(15C_2)/(15C_1) + 3. (15C_3)/(15C_2) +……..+ 15. (15C_15)/(15C_14) = 120

Prove that C_(0)-(1)/(3)*C_(1)+(1)/(5)*C_(2) - …+(-1)^(n)*(1)/(2n+1)C_(n) =(2^(2n)(n !)^(2))/((2n+1)!)

AAKASH SERIES-BINOMIAL THEOREM-ADDITIONAL EXERCISE
  1. Find the number of distinct terms in the expansion (x+ y + z)^10+ ...

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  2. Find the coefficient of x^10 in the expansion of (1+2x)^21 + (1 + 2x)^...

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  3. The expansion [x+(x^3-1)^(1//2)]^5+[x-(x^3-1)^(1//2)]^5 is a polynomia...

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  4. If the middle term of the expansion of (1+x)^(2n) is the greatest term...

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  5. Show that the greatest coefficient in the expansion of ( x + y + z + w...

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  6. Find the sum of the coefficients of integral powers of x in (1+3sqrtx)...

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  7. If n is integer greater than 1 show that a -^n C1 (a - 1) +^n C2(a...

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  8. Prove that (C0)/(1)+ (C2)/(3) + (C4)/(5) + (C6)/(7) +…….= (2^n)/(n+ 1)

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  9. Prove that (1)/(1!(n-1)!) + (1)/(3!(n-3)!)+ (1)/(5!(n-5)!) + …….= ...

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  10. Show that sum(r = 0)^50 (""^100Cr).(""^200C(150+r)) = ""^300C50

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  11. Show that C0 + (C0 + C1) + (C0 +C1 + C2)+……. + (C0 + C1 + …+Cn)= (n+2...

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  12. If x > 0 , write the first negative term in the expansion of (1 + 2x)^...

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  13. Find the coefficient of x^n in the expansion of (1+ 2x + 3x^2 + 4x^3 +...

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  14. Find the coefficient of x^6 in (1+ x + x^2 + x^3+…….oo)(1+x^2 +x^4 + x...

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  15. Find the coefficient of x^5 in (1 + x + x^2 + x^3)^5 (1+x+x^2 + x^3 + ...

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  16. Find the coefficient of x^7 in the expansion of (1+2x + 3x^2 + 4x^3 + ...

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  17. If x is small and if the expansion of a + (b)/(1 + 2x) + ( c)/(1 -3x^2...

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  18. If x nearly equal to 1 show that (px^p - qx^q)/(p-q) = x^(p+q) (nearl...

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  19. If x nearly equal to 1 show that (px^q - qx^p)/(x^q - x^p) = (1)/(1 -...

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  20. 1+n. (2n)/(1+n)+(n(n+1))/(1.2) ((2n)/(1+n))^2+…..

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