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Prove that overset(n)underset(r=0)(Sigm...

Prove that `overset(n)underset(r=0)(Sigma^(n))C_(r).4^(r)=5^(n)`

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The correct Answer is:
N/a
Binomial theorem can be written as
`overset(n)underset(r=0)(Sigma).^(n)C_(r).(1)^(n-r)x^(r)=(1+x)^(n)`
`rArr " "overset(n)underset(r=0)(Sigma).^(n)C_(r).x^(r)=(1+x)^(n)`
Putting x=4
`overset(n)underset(r=0)(Sigma).^(n)C_(r).4^(r)=(1+4)^(n)`
`=5^(n)`. Hence proved.
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NAGEEN PRAKASHAN ENGLISH-BINOMIAL THEOREM-Miscellaneous Exericse
  1. Prove that overset(n)underset(r=0)(Sigma^(n))C(r).4^(r)=5^(n)

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  2. Find a, b and n in the expansion of (a+b)^nif the first three terms ...

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  3. If the coefficients of x^2a n d\ x^3 in the expansion o (3+a x)^9 are ...

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  4. Find the coefficient of a^4 in the product (1+a)^4(2-a)^5 using binomi...

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  5. If a and b are distinct integers, prove that a - b is a factor of a^n-...

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  6. Evaluate (sqrt(3)+sqrt(2))^6-(sqrt(3)-sqrt(2))^6dot

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  7. Find the value of (a^2+sqrt(a^2-1))^4+(a^2-sqrt(a^2-1))^4.

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  8. Find an approximation of (0. 99)^5 using the first three terms of its ...

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  9. Find n, if the ratio of the fifth term from the beginning to the fi...

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  10. Using binomial theorem expand (1+x/2-2/x)^4,\ x!=0.

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  11. Find the expansion of (3x^2-2a x+3a^2)^3 using binomial theorem.

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  12. Find a, b and n in the expansion of (a+b)^nif the first three terms ...

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  13. If the coefficients of x^2a n d\ x^3 in the expansion o (3+a x)^9 are ...

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  14. Find the coefficient of x^5 in the expansion of (1 + 2x)^6 (1-x)^7.

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  15. If a and b are distinct integers, prove that a - b is a factor of a^n-...

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  16. Evaluate (sqrt(3)+sqrt(2))^6-(sqrt(3)-sqrt(2))^6dot

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  17. Find the value of (a^2+sqrt(a^2-1))^4+(a^2-sqrt(a^2-1))^4.

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  18. Find an approximation of (0. 99)^5using the first three terms of its ...

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  19. Find n, if the ratio of the fifth term from the beginning to the fi...

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  20. Expand using Binomial Theorem (1+x/2-2/x)^4,x!=0.

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  21. Find the expansion of (3x^2-2a x+3a^2)^3 using binomial theorem.

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