coefficient is n C r – . Similarly, the coefficients of r th and ( r + ) th terms are n C r – and n C r , respectively. Since the coefficients are in the ratio : : , so we have, C C , i.e., n – r + = ... ( ) and C C , i.e., n – r + = ...
( ) Solving equations( ) and ( ), we get, n = . BINOMIAL THEOREM EXERCISE . Find the coefficient of . x in ( x + ) .
a b in ( a – b ) . Write the general term in the expansion of . ( x – y ) . ( x – yx ) , x ≠ .
. Find the th term in the expansion of ( x – y ) . . Find the th term in the expansion of , x ≠ .
Find the middle terms in the expansions of . − x . . .
In the expansion of ( + a ) m + n , prove that coefficients of a m and a n are equal. . The coefficients of the ( r – ) th , r th and ( r + ) th terms in the expansion of ( x + ) n are in the ratio : : . Find n and r .
. Prove that the coefficient of x n in the expansion of ( + x ) n is twice the coefficient of x n in the expansion of ( + x ) n – . . Find a positive value of m for which the coefficient of x in the expansion ( + x ) m is .
Miscellaneous Examples Example Find the term independent of x in the expansion of .