Find the middle term or terms in the expansion of: x 3 + 9 y 10 .

Given, x 3 + 9 y 10 Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of x 3 + 9 y 10 is 10 2 + 1 t h term = 6 t h term. ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 6 t h term in the expansion of x 3 + 9 y 10 = C 5 10 x 3 5 9 y 5 = 10 ! 515 ! x 5 3 5 · 3 2 5 y 5 ∵ C r n = n ! r ! ( n - r ) ! = 10 × 9 × 8 × 7 × 6 5 × 4 × 3 × 2 × 1 · x 5 y 5 · 3 5 = 61236 x 5 y 5 ∴ Middle term in the expansion of x 3 + 9 y 10 = 61236 x 5 y 5 .

Find the middle term or terms in the expansion of: 2 a x - b x 2 12 .

Given, 2 a x - b x 2 12 Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of 2 a x - b x 2 12 = 12 2 + 1 t h = 7 t h term. ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 7 t h term in the expansion of 2 a x - b x 2 12 = 12 C 6 2 a x 6 - b x 2 6 = 12 ! 6 ! 6 ! 2 6 a 6 x 6 b 6 x 12 = 12 × 11 × 10 × 9 × 8 × 7 × 6 ! 6 × 5 × 4 × 3 × 2 × 1 × 6 ! 2 6 a 6 b 6 x 6 = 59136 a 6 b 6 x 6 ∴ Middle term in the expansion of 2 a x - b x 2 12 = 59136 a 6 b 6 x 6 .

Find the middle term or terms in the expansion of : 3 x - x 3 6 9 .

Given, 3 x - x 3 6 9 Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of 3 x - x 3 6 9 = 5 t h and 6 t h term. ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 5 t h term in the expansion of 3 x - x 3 6 9 = C 4 9 3 x 5 - x 3 6 4 = 9 ! 4 ! 5 ! 3 5 · x 5 · x 12 2 4 · 3 4 ∵ C r n = n ! r ! n - r ! = 9 × 8 × 7 × 6 × 5 ! 4 × 3 × 2 × 1 × 5 ! · 3 16 · x 17 ∵ n ! = n ( n - 1 ) ! = 189 8 x 17 ∴ 5 t h term in the expansion of 3 x - x 3 6 9 = 189 8 x 17 . Now, 6 t h term in the expansion of 3 x - x 3 6 9 = C 5 9 3 x 4 - x 3 6 5 = - C 5 9 3 4 · x 4 · x 15 2 5 · 3 5 = - 9 ! 5 ! 4 ! x 19 2 5 · 3 ∵ C r n = n ! r ! n - r ! = - 9 × 8 × 7 × 6 × 5 ! 5 ! × 4 × 3 × 2 × 1 · x 19 2 5 . 3 ∵ n ! = n ( n - 1 ) ! = - 21 16 x 19 ∴ 6 t h term in the expansion of 3 x - x 3 6 9 = - 21 16 x 19 . ∴ Middle terms in the expansion of 3 x - x 3 6 9 are 189 8 x 17 and - 21 16 x 19 .

Show that the middle term in the expansion of x - 1 x 2 n is 1 . 3 . 5 … ( 2 n - 1 ) n ! . - 2 n .

Given, x - 1 x 2 n Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of x - 1 x 2 n = 2 n 2 + 1 t h term = ( n + 1 ) t h term. ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ ( n + 1 ) t h term in the expansion of x - 1 x 2 n = C n 2 n x n - 1 x n = 2 n ! n ! n ! - 1 n ∵ C r n = n ! r ! n - r ! = ( 2 n ) ( 2 n - 1 ) ( 2 n - 2 ) ( 2 n - 3 ) ( 2 n - 4 ) … . . 6 · 5 · 4 · 3 · 2 · 1 n ( n - 1 ) ( n - 2 ) . . . . 3 · 2 · 1 - 1 n n ! ∵ n ! = n ( n - 1 ) ( n - 2 ) . . . . 3 · 2 · 1 = ( - 2 ) n ( 2 n - 1 ) ( 2 n - 3 ) . . . . 5 · 3 · 1 n ! ∴ Middle term in the expansion of x - 1 x 2 n = 1 . 3 . 5 … ( 2 n - 1 ) n ! . - 2 n . Hence, proved.