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
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 :
= 6^{th} 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 6^{th} term, which is the middle term.
T_{6} = 252x^{5}(32)
T_{6} = 8064x^{5}
Problem 2 :
In the epansion of the following binomial
(a + b)^{2k - 4}
if 8^{th} 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 = 8^{th} term
k - 1 = 8
k = 9
Problem 3 :
Find the middle term in the expansion of the following binomial.
Solution :
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