**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)} = ^{n}c_{r} x^{(n-r)} a^{r}

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 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}

**Example 1 :**

Find the middle term in the expansion of (3x - 2x^{2}/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)} = ^{n}c_{r} x^{(n-r)} a^{r}**

x = 3x, a = 2x^{2}/3, r = 4 and n = 8

T _{(4 + 1)} = ^{8}c_{4} (3x)^{(8-4)} (2x^{2}/3)^{4}

= (8 **⋅** 7 **⋅** 6 **⋅** 5)/ (4 **⋅** 3 **⋅** 2 **⋅** 1)(3x)^{4} (2x^{2}/3)^{4}

= (8 **⋅** 7 **⋅** 6 **⋅ **5)/ (4 **⋅** 3 **⋅** 2 **⋅** 1)(3x)^{4} (2x^{2}/3)^{4}

= 70(81x^{4})(16x^{8}/81)

= 70(16)x^{12}

= 1120 x^{12}

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)} = ^{n}c_{r} x^{(n-r)} a^{r}**

x = b/x, a = x/b, r = 8 and n = 16

T _{(8 + 1)} = ^{16}c_{8} (b/x)^{(16-8)} (x/b)^{8}

= ^{16}c_{8} (b/x)^{8} (x/b)^{8}

= ^{16}c_{8}

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)} = ^{n}c_{r} x^{(n-r)} a^{r}**

x = a/x, a = - √x, r = 8 and n = 16

T _{(8 + 1)} = ^{16}c_{8} (a/x)^{(16-8)} (- √x)^{8}

= ^{16}c_{8} a^{8} x^{-8 }

= ^{16}c_{8} a^{8} x^{-4}

= ^{16}c_{8} a^{8}/x^{4}

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)} = ^{n}c_{r} x^{(n-r)} a^{r}**

x = x, a = -2y, r = 6 and n = 13

T _{(6 + 1)} = ^{13}c_{6} (x)^{(13-6)} (-2y)^{6}

= ^{13}c_{6} x^{7} (-2)^{6 }y^{6}

= ^{13}c_{6} x^{7} 2^{6 }y^{6}

T_{(n + 3)/2 = } T _{(13+3)/2 ==> }T _{8}

x = x, a = -2y, r = 7 and n = 13

T _{(7 + 1)} = ^{13}c_{6} (x)^{(13-7)} (-2y)^{7}

= ^{13}c_{6} x^{6} (-2)^{7 }y^{7}

= - ^{13}c_{6} x^{6} 2^{7 }y^{7}

**Example 5 :**

Find the middle term in the expansion of (x + 2/x^{2})^{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)} = ^{n}c_{r} x^{(n-r)} a^{r}**

x = x, a = 2/x^{2}, r = 8 and n = 17

T _{(8 + 1)} = ^{17}c_{8} (x)^{(17-8)} (2/x^{2})^{8}

= ^{17}c_{8} x^{9} (2)^{8 }x^{-16}

= ^{17}c_{8} x^{9-16} 2^{8}

= ^{17}c_{8} x^{-7} 2^{8}

= ^{17}c_{8} (2^{8}/x^{7})

T_{(n + 3)/2 = } T _{(17+3)/2 ==> }T _{10}

x = x, a = 2/x^{2}, r = 9 and n = 17

T _{(9 + 1)} = ^{17}c_{9} (x)^{(17-9)} (2/x^{2})^{9}

= ^{17}c_{9} (x)^{8 }(2^{9}/x^{18})

= ^{17}c_{9} (x)^{8 }(2^{9}x^{-18})

= ^{17}c_{9} (x)^{8-18 }2^{9}

= ^{17}c_{9} x^{-10 }2^{9}

= ^{17}c_{9} (2^{9}/x^{10})

After having gone through the stuff given above, we hope that the students would have understood, "How to find middle term of an expansion".

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