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

Prove that `(.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1)`.

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Prove that .^(n)C_(0) + (.^(n)C_(1))/(2) + (.^(n)C_(2))/(3) + "……" +(. ^(n)C_(n))/(n+1) = (2^(n+1)-1)/(n+1) .

Prove that .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "…." + n xx .^(n)C_(n) = n2^(n-1) . Hence, prove that .^(n)C_(1).(.^(n)C_(2))^(2).(.^(n)C_(3))^(3)"......."(.^(n)C_(n))^(n) le ((2^(n))/(n+1))^(.^(n+1)C_(2)) AA n in N .

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2n+1) 2^(n) .

Prove that .^(n)C_(1) - (1+1/2) .^(n)C_(2) + (1+1/2+1/3) .^(n)C_(3) + "…" + (-1)^(n-1) (1+1/2+1/3 + "…." + 1/n) .^(n)C_(n) = 1/n

The value of (.^(n)C_(0))/(n)+(.^(n)C_(1))/(n+1)+(.^(n)C_(2))/(n+2)+"..."+(.^(n)C_(n))/(2n)

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that if ngt7 then .^(n-1)C_(3)+.^(n-1)C_(4)gt^(n)C_(3)

Find the sum .^(n)C_(1) + 2 xx .^(n)C_(2) + 3 xx .^(n)C_(3) + "……" + n xx .^(n)C_(n) .

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

Prove that , .^(2n)C_(n)=2^(n)(1.3.5...(2n-1))/(lfloorn)

CENGAGE PUBLICATION-BINOMIAL THEOREM-All Questions
  1. If x+y=1, prove that sum(r=0)^n .^nCr x^r y^(n-r) = 1.

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  2. Find the sum 3 C1 + 5C2

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

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  4. 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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  5. If underset(r=0)overset(2n)suma(r) (x-2)^(r) = underset(r=0)overset(2n...

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  6. 3^(2n+2)-8n-9 is divisible by

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

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

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  9. The value of "^30C0.^30C10+^30C1.^30C11+^30C2.^30C(12)+……+^30C(20) . ^...

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  10. If f(x)=x^n ,f(1)+(f^1(1))/1+(f^2(1))/(2!)+........(f^n(1))/(n !),w h ...

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  11. The fractional part of =(2^(4n))/15 is

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  12. The value of .^(15)C(0)^(2)-.^(15)C(1)^(2)+.^(15)C(2)^(2)-"...."-.^(15...

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  13. If the sum of the coefficients in the expansion of (1-3x+10 x^2)^ni sa...

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  14. If (1+x-2x^2)^6=1+a1x+a2x^2+……+a(12)x^12 then

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  15. Maximum sum of coefficient in the expansion of (1-xsintheta+x^2)^n is

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  16. If the sum of the coefficients in the expansion of (a+b)^(n) is 4096, ...

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  17. The number of distinct terms in the expansion of (x+1/x+x^2+1/(x^2))^(...

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  18. The sum of the coefficients of even power of x in the expansion of (1+...

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  19. Second term of a GP is 6 and its 5th term is 9th time of its 3rd term....

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  20. The 4th term and the 9th term of a GP is 54 and 13122 respectively. Fi...

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