HOW TO FIND THE MIDDLE TERM OF A BINOMIAL EXPANSION

Consider the following binomial with the exponent n.

(a + b)n

Then the formula to find the middle term in the expansion of the above binomial is

Key Concept

Solved Problems

Problem 1 :

Find the middle term of the following binomial expansion.

(x + 2)10

Solution :

The given binomial (x + 2)10 has the exponent 10. 

Then the number of terms in the expansion :

= 10 + 1

= 11

Middle Term :

= 6th term

Formula to find (r + 1)th term in the epansion of (a + b)n :

Comparing (a + b)n and (x + 3)10, we get

a = x, b = 2 and n = 10

Substitute the above values into the formula.

Substitute r = 5 to get the 6th term, which is the middle term.

T6 = 252x5(32)

T6 = 8064x5

Problem 2 :

In the epansion of the following binomial 

(a + b)2k - 4

if 8th term is the middle term, then find the value of k.

Solution :

The exponent of the given binomial is (2k - 4).

Then the number of terms in the expansion :

= (2k - 4) + 1

= 2k - 4 + 1

= 2k - 3

Middle term = 8th term

k - 1 = 8

k = 9

Problem 3 :

Find the middle term in the expansion of the following binomial.

Solution :

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