(i) (x/3 + 9y)10
Here we have,
(x/3 + 9y)10 where, n = 10 is an even number.
Therefore the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. i.e., 6th term.
Now,
T6 = T5+1
Thus, the middle term is 61236x5y5.
(ii) (3 – x3/6)7
Here we have,
(3 – x3/6)7 where, n = 7 (odd number).
Therefore the middle terms are ((n + 1)/2) = ((7 + 1)/2) = 8/2 = 4 and
((n + 1)/2 + 1) = ((7 + 1)/2 + 1) = (8/2 + 1) = (4 + 1) = 5
The terms are 4th and 5th.
Now,
T4 = T3+1
= 7C3 (3)7-3 (-x3/6)3
= -105/8 x9
And,
T5 = T4+1
= 9C4 (3)9-4 (-x3/6)4
Hence, the middle terms are -105/8 x9 and 35/48 x12.
(iii) (2ax – b/x2)12
We have,
(2ax – b/x2)12 where, n = 12 is an even number.
Therefore the middle term is (n/2 + 1) = (12/2 + 1) = (6 + 1) = 7. i.e., 7th term.
Now,
T7 = T6+1
Thus, the middle term is (59136a6b6)/x6.
(iv) (p/x + x/p)9
We have,
(p/x + x/p)9 where, n = 9 (odd number).
Therefore the middle terms are ((n + 1)/2) = ((9 + 1)/2) = 10/2 = 5 and
((n + 1)/2 + 1) = ((9 + 1)/2 + 1) = (10/2 + 1) = (5 + 1) = 6
The terms are 5th and 6th.
Now,
T5 = T4+1
And,
T6 = T5+1
= 9C5 (p/x)9-5 (x/p)5
Thus, the middle terms are 126p/x and 126x/p.
(v) (x/a – a/x)10
We have,
(x/a – a/x) 10 where, n = 10 (even number)
Therefore the middle term is (n/2 + 1) = (10/2 + 1) = (5 + 1) = 6. ie., 6th term
Now,
T6 = T5+1
Thus, the middle term is -252.