In this section, you will learn how to find the middle term of an expansion.
To find a particular term of an expansion, we can use the formula given below.
T(r+1) = ncr x(n-r) ar
The number of terms in the expansion of (x + a)n depends upon the index n. The index is either even (or) odd.
Let us see how to find the middle term.
Case (i) : n is even
The number of terms in the expansion is (n + 1), which is odd. Hence, there is only one middle term and it is given by T(n/2) + 1
Case (ii) : n is odd
The number of terms in the expansion is (n + 1), which is even. Hence, there are two middle terms and they are given by T(n + 1)/2 and T(n + 3)/2
Example 1 :
Find the middle term in the expansion of (3x - 2x2/3)8.
Solution :
Here n = 8, that is even
So, the middle term = T(n/2) + 1
= T (8/2) + 1
= T (4 + 1) ==> T 5
General term :
T(r+1) = ncr x(n-r) ar
x = 3x, a = 2x2/3, r = 4 and n = 8
T (4 + 1) = 8c4 (3x)(8-4) (2x2/3)4
= (8 ⋅ 7 ⋅ 6 ⋅ 5)/ (4 ⋅ 3 ⋅ 2 ⋅ 1)(3x)4 (2x2/3)4
= (8 ⋅ 7 ⋅ 6 ⋅ 5)/ (4 ⋅ 3 ⋅ 2 ⋅ 1)(3x)4 (2x2/3)4
= 70(81x4)(16x8/81)
= 70(16)x12
= 1120 x12
Example 2 :
Find the middle term in the expansion of (b/x - x/b)16.
Solution :
Here n = 16, that is even
So, the middle term = T(n/2) + 1
= T (16/2) + 1
= T (8 + 1) ==> T 9
General term :
T(r+1) = ncr x(n-r) ar
x = b/x, a = x/b, r = 8 and n = 16
T (8 + 1) = 16c8 (b/x)(16-8) (x/b)8
= 16c8 (b/x)8 (x/b)8
= 16c8
Example 3 :
Find the middle term in the expansion of (a/x - √x)16.
Solution :
Here n = 16, that is even
So, the middle term = T(n/2) + 1
= T (16/2) + 1
= T (8 + 1) ==> T 9
General term :
T(r+1) = ncr x(n-r) ar
x = a/x, a = - √x, r = 8 and n = 16
T (8 + 1) = 16c8 (a/x)(16-8) (- √x)8
= 16c8 a8 x-8
= 16c8 a8 x-4
= 16c8 a8/x4
Example 4 :
Find the middle term in the expansion of (x - 2y)13.
Solution :
Here n = 13, that is even
So, the middle term = T(n + 1)/2 and T(n + 3)/2
T(n + 1)/2 = T (13+1)/2 ==> T 7
General term :
T(r+1) = ncr x(n-r) ar
x = x, a = -2y, r = 6 and n = 13
T (6 + 1) = 13c6 (x)(13-6) (-2y)6
= 13c6 x7 (-2)6 y6
= 13c6 x7 26 y6
T(n + 3)/2 = T (13+3)/2 ==> T 8
x = x, a = -2y, r = 7 and n = 13
T (7 + 1) = 13c6 (x)(13-7) (-2y)7
= 13c6 x6 (-2)7 y7
= - 13c6 x6 27 y7
Example 5 :
Find the middle term in the expansion of (x + 2/x2)17.
Solution :
Here n = 17, that is even
So, the middle term = T(n + 1)/2 and T(n + 3)/2
T(n + 1)/2 = T (17+1)/2 ==> T 9
General term :
T(r+1) = ncr x(n-r) ar
x = x, a = 2/x2, r = 8 and n = 17
T (8 + 1) = 17c8 (x)(17-8) (2/x2)8
= 17c8 x9 (2)8 x-16
= 17c8 x9-16 28
= 17c8 x-7 28
= 17c8 (28/x7)
T(n + 3)/2 = T (17+3)/2 ==> T 10
x = x, a = 2/x2, r = 9 and n = 17
T (9 + 1) = 17c9 (x)(17-9) (2/x2)9
= 17c9 (x)8 (29/x18)
= 17c9 (x)8 (29x-18)
= 17c9 (x)8-18 29
= 17c9 x-10 29
= 17c9 (29/x10)
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