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Show that, 3^(n)-(n)/(1!)3^(n-1)+(n(n-...

Show that,
`3^(n)-(n)/(1!)3^(n-1)+(n(n-1))/(2!)3^(n-2)-...+(-1)^(n)=""^(n)C_(0)+""^(n)C_(1)+""^(n)C_(2)+...+ ""^(n)C_(2)`

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Explore conceptually related problems

The value of (""^(n) C_(1)+2. ""^(n) C_(2)+3. ""^(n)C_(3)+…+ n""^(n)C_(n)) is-

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

Show that , (.^(n)C_(r)+^(n)C_(r-1))/(.^(n)C_(r-1)+^(n)C_(r-2))=(.^(n+1)p_(r))/(r.^(n+1)p_(r-1))

Evaluate .^(n)C_(0).^(n)C_(2)+.^(n)C_(1).^(n)C_(3)+.^(n)C_(2).^(n)C_(4)+"...."+.^(n)C_(n-2).^(n)C_(n) .

Show that : (^(4n)C_(2n))/(^(2n)C_n) = (1.3.5...(4n-1))/{1.3.5...(2n-1)}^2

Show that .^(n)C_(r)+.^(n-1)C_(r-1)+.^(n-1)C_(r-2)=.^(n+1)C_(r) .

If n is a positive integer then using the indentiy (1+x)^(n)=(1+x)^(3)(1+x)^(n-3) , prove that ""^(n)C_(r)=""^(n-2)C_(r)+3*""^(n-3)C_(r-1)*""^(n-3)C_(r-2)+""^(n-3)C_(r-3)

Find the following sums : (i) .^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....." (ii) .^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...." (iii) .^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....." (iv) .^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......" (v) .^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...." (vi) .^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."

Prove that , .^(n)C_(r)+3.^(n)C_(r-1)+3.^(n)C_(r-2)+^(n)C_(r-3)=^(n+3)C_(r)

If (1+x)^(n)=1+""^(n)C_(1)x+""^(n)C_(2)x^(2)+….+""^(n)C_(n) x^(n) show that, n*2^(n-1)=""^(n)C_(1)+2*""^(n)C_(2)+….. +n*""^(n)C_(n) .

CHHAYA PUBLICATION-MISCELLANEOUS EXAMPLES-ALGEBRA
  1. If z(1)+z(2) are two complex number and |(barz(1)-2bar z(2))/(2-z(1)ba...

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  2. If x+iy moves on the line 3x+4y+5=0, prove that the minimum value of ...

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  3. If |z^(2)|=9|z|, find the maximum and minimum values of |z|.

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  4. Find the greatest value of |z| when |z-(6)/(z)|=2,z being a complex nu...

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  5. Find the least value of |z+(1)/(z)| if |z| ge 3 (z being a complex ...

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  6. If the roots of the equation ix^(2)-(b^(2)+c^(2))x+(c^(2)+d^(2))^(2)=0...

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  7. If one root of the equation ax^(2)-2ibx-ic=0 be square of the other, t...

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  8. Solve (7y^(2)+1)/(y^(2)-1)-4 ((y^(2)-1)/(7y^(2)+1))=-3

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  9. Solve (x-(x)/(x+1))^(2)+2x((x)/(x+1))=3

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  10. If alpha and beta are the roots of ax^(2)+bx+cid=0 find the equation ...

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  11. Show that, 3^(n)-(n)/(1!)3^(n-1)+(n(n-1))/(2!)3^(n-2)-...+(-1)^(n)="...

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  12. Find the coefficient of x^(-5) in ""^(n)C(0)-""^(n)C(1) ((2x-1)/(x)...

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  13. Find the degree of theexpansion, {x+(x^(3)-1)^(1//2)}^(5)+{x-(x^(3)-...

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  14. Find the term independent of x in the expansion, ((x+1)/(x^(2//3)-x^(...

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  15. Find the remainder if number 2^(2003) is divided by 17.

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  16. Find last three digits of the number 17^(256).

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  17. Find the sum of the series, (1^(2))/(3)+(1^(2)+2^(2))/(5)+(1^(2)+2^(...

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  18. Find the value of sum(r=1)^(n) (r)/(1+r^(2)+r^(4))

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  19. If a(1),a(2),a(3),.. from a G.P. with common ratio r, find in terms ...

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  20. Show that (111...1)/(91 "digits") is a composite number.

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