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Prove that (1^2)/3""^n C1+(1^2+2^2)/5""^...

Prove that `(1^2)/3""^n C_1+(1^2+2^2)/5""^n C_2+(1^1+2^2+3^2)/7""^n C_3+... +(1^2+2^2++n^2)/(2n+1)""^n C_n`=`((n(n+3))/6)*2^(n-2)dot`

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CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. Find the sum sumsum(0lt=i<jlt=n-1)^n Cidot

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

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

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  5. Find the sum 2..^(10)C(0) + (2^(2))/(2).^(10)C(1) + (2^(3))/(3).^(1...

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  6. If in the expansion of (2x+5)^(10) , the numerically greatest term in ...

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  7. Find the value of (1)/(81^(n))-(10)/(81^(n)).^(2n)C(1) + (10^(2))/(8...

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  8. Find the value of 5C3 + 4C2

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  9. Find the sum .^1C0+^2C1+^3C2+....+^(n+1)Cn ,where Cr=^n Crdot

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

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

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

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

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  14. Find the sum 3 C1 + 5C2

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

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

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  18. 3^(2n+2)-8n-9 is divisible by

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

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

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