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If k and n are positive integers and S(...

If k and n are positive integers and `S_(k) = 1^(k) + 2^(k) + 3^(k) + "……" + n^(k)`, then prove that `underset(r=1)overset(m)sum.^(m+1)C_(r )s_(r) = (n+1)^(m+1) - (n+1)`

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CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. Find the sum sum(i=0)^r.^(n1)C(r-i) .^(n2)Ci .

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  2. Prove that sum(r=0)^(2n)(r. ^(2n)Cr)^2=n^(4n)C(2n) .

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  3. If k and n are positive integers and S(k) = 1^(k) + 2^(k) + 3^(k) + "...

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  4. Prove that sum(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)(.^n Cr)=1/n .

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  5. Prove that (C1)/1-(C2)/2+(C3)/3-(C4)/4+....+((-1)^(n-1))/n Cn=1+1/2+1/...

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  6. Prove that underset(r=0)overset(n)sum .^(n)C(r )sin rx. cos (n-r)x = 2...

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  7. Find the last two digits of the number (23)^(14)dot

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  8. Find the last two digits of the number 27^(27)dot

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  9. Find the number of nonzero terms in the expansion of (1+3sqrt(2)x)^9+(...

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  10. Find the value of (sqrt(2)+1)^6-(sqrt(2)-1)^6dot

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  11. Using the binomial theorem (without using the formula for .^(n)C(r)), ...

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  12. Find the value of .^(4n)C0+^(4n)C4+^(4n)C8+...+""^(4n)C(4n) .

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  13. Find the degree of the polynomial 1/(sqrt(4x+1)){((1+sqrt(4x+1))/2)^7-...

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  14. Find the coefficient of x^(n) in the polynomial (x+.^(n)C(0)) (x+3.^(n...

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  15. If (1+x)^(15)=C0+C1x+C2x^2++C(15)x^(15), then find the sum of C1+2C3+3...

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

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  17. Find the sum sumsum(0lt=i<jlt=n-1)^n Cidot

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  18. Show that the integer next above (sqrt(3)+1)^(2m) contains 2^(m+1), as...

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  19. Prove that (1^2)/3""^n C1+(1^2+2^2)/5""^n C2+(1^1+2^2+3^2)/7""^n C3+.....

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  20. Prove that (1)/(n+1)=(.^(n)C(1))/(2)-(2(.^(n)C(2)))/(3)+(3(.^(n)C(3)...

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