Home
Class 11
MATHS
In the expansion of (1+a)^(m+n) ,prove t...

In the expansion of `(1+a)^(m+n)` ,prove that coefficients of `a^(m)` and `a^(n)` are equal.

Text Solution

Verified by Experts

The correct Answer is:
N/a
`T_(r+1)=^(m+n)C_(r)(1)^(m+n-r).a^(r)`
`=^(m+1)C_(r).a^(r)`
`:.` Coefficient of `a^(m)=^(m+m)C_(m)=(|ul(M+n))/(|ulm.|uln)(" Put "r=m)`
and coefficient of `a^(n)=^(m+n)C_(n)=(|ul(m+n))/(|uln.|ulm)(" Put " r=n)` ltbr. Therefore coefficient of `a^(m)` and coefficient of `a^(n)` are equal.
Hence Proved.
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exericse|20 Videos

  • BINOMIAL THEOREM

    NAGEEN PRAKASHAN ENGLISH|Exercise Exericse 8.1|28 Videos

  • COMPLEX NUMBERS AND QUADRATIC EQUATION

    NAGEEN PRAKASHAN ENGLISH|Exercise MISCELLANEOUS EXERCISE|20 Videos

Similar Questions

Explore conceptually related problems

In the binomial expansion of (1+ x)^(m+n) , prove that the coefficients of x^(m) and x^(n) are equal.

In the binomial expansion of (1+a)^(m+n) , prove that the coefficient of a^m a n d\ a^n are equal.

In the expansion of (3+x/2)^(n) the coefficients of x^(7) and x^(8) are equal, then the value of n is equal to

If in the expansion of (1 +x)^(m) (1 - x)^(n) , the coefficients of x and x^(2) are 3 and - 6 respectively, the value of m and n are

Statement-I : In the expansion of (1+ x)^n ifcoefficient of 31^(st) and 32^(nd) terms are equal then n = 61 Statement -II : Middle term in the expansion of (1+x)^n has greatest coefficient.

If in the expansion of (1 + x)^n , the coefficients of pth and qth terms are equal, prove that p+q=n+ 2 , where p!=q .

If m and n are positive integers, then prove that the coefficients of x^(m) " and " x^(n) are equal in the expansion of (1+x)^(m+n)

If x^p occurs in the expansion of (x^2 + (1)/(x) )^(2n) , prove that its coefficient is ((2n)!)/( [ (1)/(3) (4n - p)!][(1)/(3) (2n + p) !])

In the expansion of (x^(2) + (1)/(x) )^(n) , the coefficient of the fourth term is equal to the coefficient of the ninth term. Find n and the sixth term of the expansion.

Prove that he coefficient of x^n in the expansion of (1+x)^(2n) is twice the coefficient of x^(n) in the expansion of (1+x)^(2n-1)

NAGEEN PRAKASHAN ENGLISH-BINOMIAL THEOREM-Exericse 8.2
  1. Find the 4^(t h)term in the expansion of (x-2y)^(12).

    Text Solution

    |

  2. Find 13th term in the expansion of (9x-1/(3x))^(18),\ x!=0.

    Text Solution

    |

  3. Find the middle terms in the expansion of (3 - (x^3)/( 6) )^7.

    Text Solution

    |

  4. Find the middle term in the expansion of :\ (x/3+9y)^(10)

    Text Solution

    |

  5. In the binomial expansion of (1+a)^(m+n) , prove that the coefficient ...

    Text Solution

    |

  6. The coefficients of the (r-1)^(th),r^(th) and (r+1)^(th) terms in the ...

    Text Solution

    |

  7. The coefficient of x^(n) in the expansion of (1 + x)^(2n) " and " (1 +...

    Text Solution

    |

  8. Find a positive value of m for which the coefficient of x^(2) in the e...

    Text Solution

    |

  9. Find the coefficient of x^5""in(x+3)^8

    Text Solution

    |

  10. Find the coefficient of a^5b^7in(a-2b)^(12)

    Text Solution

    |

  11. Write the general term in the expansion of (x^2-y)^6dot

    Text Solution

    |

  12. Write the general term in the expansion of (x^2-y x)^(12),x!=0

    Text Solution

    |

  13. Find the 4^(t h)term in the expansion of (x-2y)^(12).

    Text Solution

    |

  14. Find the 13^(t h)term in the expansion of (9x-1/(3sqrt(x)))^(18),x!=0

    Text Solution

    |

  15. Find the middle term in the expansion of (3-(x^(3))/(6))^(7)

    Text Solution

    |

  16. Find the middle term in the expansion of :\ (x/3+9y)^(10)

    Text Solution

    |

  17. In the expansion of (1+a)^(m+n) ,prove that coefficients of a^(m) and ...

    Text Solution

    |

  18. The coefficients of the (r-1)^(th),r^(th) and (r+1)^(th) terms in the ...

    Text Solution

    |

  19. prove that the coefficient of x^n in the expansion of (1+x)^(2n) is tw...

    Text Solution

    |

  20. Find a positive value of m for which the coefficient of x^2 in the ex...

    Text Solution

    |