Sample Problems on Middle Terms

Example 1: Find the middle of the following binomial expansion (x + a)⁸

Solution: 

Given expansion is (x + a)⁸

n = 8, we consider the expansion has (n + 1) terms so the above expansion has (8 + 1) i.e 9 terms 

we have T₁, T₂, T₃, T₄, T₅, T₆ , T₇, T₈, T₉.

T_r+1 = T_{n/2 + 1} = ⁿC_n/2.x^{n – n/2}.Y^n/2

T_{8/2 + 1} = ⁸C_8/2.x^8-8/2.a^8/2

T₅ = ⁸C₄.x⁴.a⁴ is the required middle term of the given binomial expansion.

Example 2: Find the middle of the following binomial expansion (x + 3y)⁶

Solution: 

Given expansion is (x + 3y)⁶

n = 6, we consider the expansion has (n + 1) terms so the above expansion has (6 + 1) i.e 7 terms

we have T₁, T₂, T₃, T₄, T₅, T₆ , T₇.

T_r+1 = T_{n/2 + 1} = ⁿC_n/2.x^{n – n/2}.Y^n/2

T_{6/2 + 1} = ⁶C_6/2.x^6-6/2.3y^6/2

T₄ = ⁶C₃.x³.3y³ is the required middle term of the given binomial expansion.

Example 3: Find the middle of the following binomial expansion (2x + 5y)⁴

Solution: 

Given expansion is  (2x – 5y)⁴

n = 4, we consider the expansion has (n + 1) terms so the above expansion has (4 + 1) i.e 5 terms

we have T₁, T₂, T₃, T₄, T₅.

T_r+1 = T_{n/2 + 1} = ⁿC_n/2.x^{n – n/2}.Y^n/2

T_{4/2 + 1} = ⁴C_4/2.2x^4-4/2.5y^4/2

T₃ = ⁴C₂.x².5y² is the required middle term of the given binomial expansion.

If n is Odd

If n is the odd number then we make it into an even number and consider (n + 1) as even and (n + 1/2), (n + 3/2) as the middle terms. In simple, if n is odd then we consider it even.

We have two middle terms if n is odd. To find the middle term:

Consider the general term of binomial expansion i.e  

• In this case, we replace “r” with the two different values
• One term is (n + 1/2) compare with (r + 1) terms we get

r + 1 = n + 1/2

r = n + 1/2 -1

r = n -1/2

• Second middle term , compare (r + 1) with (n + 3/2) we get

r +1 = n +3/2

r = n + 3/2 – 1

r = n + 1/2

The two middle terms when n is odd are (n – 1/2) and (n + 1/2).

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