# HOW TO FIND MIDDLE TERM OF AN EXPANSION

## About "How to find middle term of an expansion"

How to find middle term of an expansion :

Here we are going to see how to find the middle term of an expansion.

To find the particular term of the expansion, we need to use the formula given below.

General term :

T(r+1)ncr x(n-r) ar

The number of terms in the expansion of (x + a)depends upon the index n. The index is either even (or) odd.

Let us find the middle terms.

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

## How to find middle term of an expansion - Examples

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

Let us see the next example on "How to find middle term of an expansion".

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

(8 + 1)  =  16c8 (b/x)(16-8) (x/b)8

=  16c8 (b/x)8 (x/b)8

= 16c8

Let us see the next example on "How to find middle term of an expansion".

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

(8 + 1)  =  16c8 (a/x)(16-8) (x)8

=  16c8 a8 x-8

=  16c8 a8 x-4

=  16c8 a8/x4

Let us see the next example on "How to find middle term of an expansion".

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

(6 + 1)  =  13c6 (x)(13-6) (-2y)6

=  13c6 x7 (-2)y6

=  13c6 x7 2y6

T(n + 3)/2   =   T (13+3)/2  ==>  8

x = x, a = -2y,  r = 7 and n = 13

(7 + 1)  =  13c6 (x)(13-7) (-2y)7

=  13c6 x6 (-2)y7

= - 13c6 x6 2y7

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  ==>  9

General term :

T(r+1) =  ncr x(n-r) ar

x = x, a = 2/x2,  r = 8 and n = 17

(8 + 1)  =  17c8 (x)(17-8) (2/x2)8

=  17c8 x9 (2)x-16

=  17c8 x9-16 28

=  17c8 x-7 28

=  17c8 (28/x7)

T(n + 3)/2   =   T (17+3)/2  ==>  10

x = x, a = 2/x2,  r = 9 and n = 17

(9 + 1)  =  17c9 (x)(17-9) (2/x2)9

=  17c9 (x)(29/x18)

=  17c9 (x)(29x-18)

=  17c9 (x)8-18 29

=  17c9 x-10 29

=  17c9 (29/x10)

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