Examples on Middle Terms

Example 1: Find the middle terms of the following binomial expansion (x + a)⁹

Solution: 

Given expansion is (x + a)⁹

a = x, b = a, and n = 9

Middle terms will be (n – 1)/2 and (n + 1)/2

T_{r + 1} =T _{n – 1/2} and T_{n + 1/2}

First middle term:

T_{r + 1} = T_{n – 1/2} = ⁹C_{(n – 1)/2}.x^{9 – (n – 1)/2}.a^{(n – 1)/2}

T_{(9 – 1/2)} = ⁹C_{(9 – 1)/2}.x^{9 – (n – 1)/2}.a^{(n – 1)/2}

T₄ = ⁹C₄.x⁵.a⁴ 

Second middle term: 

T_{r + 1} = T_{n + 1/2} =  ⁹C_{(n + 1)/2}.x^{9 – (n + 1)/2}.a^{(n + 1)/2}

T_{(9 + 1)/2} = ⁹C_{(9 + 1)/2}.x^{9 – (9 + 1)/2}.a^{(9 + 1)/2}

T₅ = ⁹C₅.x⁴.a⁵

The middle terms of expansion are T₄, and T_5.

Example 2: Find the middle terms of the following binomial expansion (4a + 9b)⁷

Solution: 

Given expansion is (4a + 9b)⁷

a = 4a,b = 9b, and n = 7

middle terms will be (n -1/2) and (n + 1/2) 

T_{r + 1} =T _{n – 1/2} and T_{n + 1/2}

First middle term:

T_{r + 1} = T_{n – 1/2} = ⁷C_{(n – 1)/2}.4a^{7 – (n – 1)/2}.9b^{(n – 1)/2}

T_{(7 – 1/2)} = ⁷C_{(7 – 1)/2}.4a^{7 – (7 – 1)/2}.9b^{(7 – 1)/2}

T₃ = ⁷C₃.4a⁴.9b³

Second middle term:

T_{r + 1} = T_{(n + 1)/2} =  ⁷C_{(n + 1)/2}.4a^{7 – (n + 1)/2}.9b^{(n + 1)/2}

T_{(7 + 1)/2} =  ⁷C_{(7 + 1)/2}.4a^{7 – (7 + 1)/2}.9b^{(7 + 1)/2}

T₄ = ⁷C₄.4a³.9b⁴

The middle terms of expansion are T₃, and T₄. 

Example 3: Find the middle terms of the following binomial expansion (2x + 8y)⁵

Solution:

Given expansion is (2x + 8y)⁵

a = 2x,b = 8y, and n = 5

Middle terms will be (n – 1)/2 and (n + 1)/2 

T_{r + 1} = T _{(n – 1)/2} and T_{(n + 1)/2}

First middle term:

T_{r + 1} = T_{(n – 1)/2} = ⁵C_{(n – 1)/2}.2x^{5 – (n – 1)/2}.8y^{(n – 1)/2}

T_{(5 – 1)/2} = ⁵C_{(5 – 1)/2}.2x^{5 – (5 – 1)/2}.8y^{(5 – 1)/2}

T₂ = ⁵C₂.2x³.8y²

Second middle term:

T_{r + 1} = T_{(n + 1)/2} =  ⁵C_{(n + 1)/2}.2x^{5 – (n + 1)/2}.8y^{(n + 1)/2}

T_{(5 + 1)/2} =  ⁵C_{(5 + 1)/2}.2x^{5 – (5 + 1)/2}.8y^{(5 + 1)/2}

T₃ = ⁵C₃.2x².8y³

The middle terms of expansion are T₂, and T₃.

Binomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial “(a + b)^n“ into a sum involving terms of the form “ax^zy^c“, the exponents z and c are non-negative integers where z + c = n, and the coefficient of each term is a positive integer depending on the values of n and b. 

Example: If n = 4

(a + b)⁴ = a⁴ + 4a³y + 6a²b² + 4ab³ + b