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Prove that sum(r=0)^n^n Cr3^r=4^n...

Prove that `sum_(r=0)^n^n C_r3^r=4^n`

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To prove that \[ \sum_{r=0}^{n} \binom{n}{r} 3^r = 4^n \] we can use the Binomial Theorem. The Binomial Theorem states that: ...
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RD SHARMA-BINOMIAL THEOREM-Solved Examples And Exercises
  1. Using binomial theorem, prove that (101)^(50)> 100^(50)+99^(50)dot

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  2. Expand (x^2+2a)^5 by binomial theorem.

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  3. Prove that sum(r=0)^n^n Cr3^r=4^n

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  4. By using binomial theorem, expand: (1+x+x^2)^3 (b) (1-x+x^2)^4

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  5. If the third term in the expansion of (1/x+x^(log10x))^5 is 1000 , the...

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  6. In the binomial expansion of (1+x)^n , coefficients of the fifth, sixt...

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  7. In the binomial expansion of (a+b)^n , coefficients of the fourth and ...

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  8. Find the value of a so that the term independent of x in (sqrt(x)-k/(x...

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  9. It n is a positive integer, prove that 3^(3n)-26n-1 is divisible by ...

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  10. Using binomial theorem, prove that 3^(2n+2)-8^n-9 is divisible by 64 ...

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  11. For what value of x is the ninth term in the expansion of (3^(log3 sqr...

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  12. If the fourth term in the expansion of {sqrt(1/("""x"log^(x+1))+1/(x 1...

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  13. If the coefficient of 2nd, 3rd and 4th terms in the expansion of (1...

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  14. If the coefficient of (2r+4)th and (r-2)th terms in the expansion of ...

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  15. Find the value of alpha for which the coefficients of the middle terms...

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  16. If the term free form x in the expansion of (sqrt(x)-m/(x^(2)))^(10) i...

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  17. If the seventh term from the beginning and end in the binomial expa...

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  18. Find n in the binomial (2 3+1/(3 3))^n , if the ration 7th term from t...

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  19. Write the number of the coefficients in the expansion of (1-3x+3x^2-x^...

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  20. Write the sum of the coefficients in the expansion of (1-3x+x^2)^111.

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