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If (1+x)^n=sum(r=0)^n Cr x^r , then prov...

If `(1+x)^n=sum_(r=0)^n C_r x^r ,` then prove that `C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot` .

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CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. If the middle term in the binomial expansion of (1/x+xsinx)^(10) is eq...

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  2. Find the sum C0+3C1+3^2C2+...+3^n Cndot

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  3. If (1+x)^n=sum(r=0)^n Cr x^r , then prove that C1+2C2+3C3+....+n Cn=n2...

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  4. If T0,T1, T2, ,Tn represent the terms in the expansion of (x+a)^n , t...

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  5. If (1+x+x^2)^n=a0+a1x+a2x^2++a(2n)x^(2n), find the value of a0+a3+a6+...

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

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  7. Prove that .^n C0+^n C3+^n C6+=1/3(2^n+2cos((npi)/3)) .

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  8. Given that the 4^(th) term in the expansion of (2+3/(8)x)^(10) has the...

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  9. Find the greatest coefficient in the expansion of (1+2x//3)^(15)dot .

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  10. Find the greatest term in the expansion of sqrt(3)(1+1/(sqrt(3)))^(20)...

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  11. Find numerically the greatest term/terms in the expansion of (3-5x)^(1...

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  12. Let n be an odd natural number greater than 1. Then , find the number...

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  13. Find the remainder when 27^(40) is divided by 12.

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  14. In the expansion of (1+x)^n , 7th and 8th terms are equal. Find the ...

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  15. Find the sum sum(j=0)^n( ^(4n+1)Cj+^(4n+1)C(2n-j)) .

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  16. Show that no three consecutive binomial coefficients can be in G.P.

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  17. Find the sum sum(r=1)^n r^n (^nCr)/(^nC(r-1)) .

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  18. Show that 9^(n+1)-8n-9 is divisible by 64, where n is a positive integ...

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  19. If the 3rd, 4th. 5th and sixth term in the expansion of (x+alpha)^n ar...

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  20. Find the remainder when 7^(98) is divided by 5.

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