Find the coefficients of x 32 and x - 17 in x 4 - 1 x 3 15 .

Let x - 17 and x 32 occurs in the r + 1 th term of x 4 - 1 x 3 15 Then T r + 1 = C r 15 x 4 15 - r · - 1 x 3 r = - 1 r · C r 15 · x 60 - 7 r . Now, for x 32 , we must have 60 - 7 r = 32 ⇒ r = 4 and for x - 17 , we must have 60 - 7 r = - 17 ⇒ r = 11 . Hence, coefficients of x 32 and x - 17 are obtained by putting r = 4 and 11 respectively. Coefficients of x 32 = - 1 4 C 4 15 = 1365 Coefficients of x - 17 = - 1 11 C 11 15 = - 1365 .

Find the coefficients of x 32 and x - 17 in x 4 - 1 x 3 15 .

Let x - 17 and x 32 occurs in the r + 1 th term of x 4 - 1 x 3 15 Then T r + 1 = C r 15 x 4 15 - r · - 1 x 3 r = - 1 r · C r 15 · x 60 - 7 r . Now, for x 32 , we must have 60 - 7 r = 32 ⇒ r = 4 and for x - 17 , we must have 60 - 7 r = - 17 ⇒ r = 11 . Hence, coefficients of x 32 and x - 17 are obtained by putting r = 4 and 11 respectively. Coefficients of x 32 = - 1 4 C 4 15 = 1365 Coefficients of x - 17 = - 1 11 C 11 15 = - 1365 .

Find the two middle terms of 3 a - a 3 6 9 .

As here n = 9 , the middle terms will be 9 + 1 2 th and 9 + 1 2 + 1 th i.e. 5 th and 6 th terms. General term T r + 1 = n C r x n - r a r in x + a n ∴ T 5 = C 4 9 3 a 5 - a 3 6 4 = 189 8 a 17 . T 6 = C 5 9 3 a 4 - a 3 6 5 = - 21 16 a 19 .

Find the term independent of x in 3 2 x 2 - 1 3 x 9 .

General term in the expansion of a + x n is T r + 1 = n C r a n - r x r Let x does not occur in the r + 1 th term of 3 2 x 2 - 1 3 x 9 Then, T r + 1 = C r 9 3 2 x 2 9 - r · - 1 3 x r = C r 9 3 2 9 - r · x 18 - 3 r · - 1 3 r For the term independent of x i.e. involving x 0 , we must have 18 - 3 r = 0 ⇒ r = 6 ∴ Hence, the 7 th term is independent of x and is C 6 9 3 2 3 - 1 3 6 = 7 18 .

Find the 13 th term of 9 x - 1 3 x 18 .

For general term, we have T r + 1 = C r n x n - r · a r in the expansion of x + a n In the expansion of 9 x - 1 3 x 18 , we have 13 th term T 12 + 1 = C 12 18 9 x 6 - 1 3 x 12 = C 12 18 9 x 6 × 1 ( 9 x ) 6 = C 12 18 = 18564 .

If x r occurs in the expansion of x + 1 x n , find its coefficient.

General term in the expansion of a + x n is T r + 1 = n C r a n - r x r Let x r occurs in the ( p + 1 ) th term of x + 1 x n , then T p + 1 = C p n x n - p · 1 x p = n C p x n - 2 p For x r , we must have n - 2 p = r ⇒ p = n - r 2 ∴ Required. coefficient of x r = n C n - r 2 = n ! n - r 2 ! × n + r 2 ! .

Find the term independent of x in x - 1 x 2 3 n .

General term in the expansion of a + x n is T r + 1 = n C r a n - r x r Let the term independent of x ( having x 0 ) in the expansion of x - 1 x 2 3 n is r + 1 th term. ∴ T r + 1 = C r 3 n x 3 n - r - 1 x 2 r = C r 3 n - 1 r x 3 n - 3 r for x 0 , we have 3 n - 3 r = 0 ⇒ r = n . Hence, required coefficient = 3 n C n - 1 n = - 1 n × 3 n ! n ! × 2 n ! .

If x p occurs in the expansion of x 2 + 1 x 2 n , prove that its coefficient is 2 n ! 1 3 4 n - p ! 1 3 2 n + p ! .

General term in the expansion of a + x n is T r + 1 = n C r a n - r x r Let x p occurs in the r + 1 th term of x 2 + 1 x 2 n Then, T r + 1 = C r 2 n x 2 2 n - r · 1 x r = 2 n C r · x 4 n - 3 r For x r , we must have 4 n - 3 r = p i.e., r = 4 n - p 3 Hence, the required coefficient = 2 n C 4 n - p 3 = 2 n ! 4 n - p 3 ! × 2 n + p 3 ! .