third term from the end is the ( n – ) th term of the expansion and n – = ( n + ) – ( – ) and so on. Thus r th term from the end will be term number ( n + ) – ( r – ) = ( n – r + ) of the expansion. And the ( n – r + ) th term is n C n – r + x r – a n – r + . Example Find the term independent of x in the expansion of , x > .
Solution We have T r + = C − C .x C r .x Since we have to find a term independent of x , i.e., term not having x , so take We get r = . The required term is C Example The sum of the coefficients of the first three terms in the expansion of m , x ≠ , m being a natural number, is . Find the term of the expansion containing x . Solution The coefficients of the first three terms of m are m C , (– ) m C and m C .
Therefore, by the given condition, we have m C – m C + m C = , i.e., – m + ) m m − BINOMIAL THEOREM which gives m = ( m being a natural number). Now T r + = C r x – r = C r (– ) r . x – r Since we need the term containing x , so put – r = i.e., r = . Thus, the required term is C (– ) x , i.e., – x .
Example If the coefficients of