find the middle term inthe expansion of (x+1/x)4
Answers
Step-by-step explanation:
The given expression is
(x-\frac{1}{x})^{10}(x−
x
1
)
10
According to the binomial expansion,
(a+b)^n=^nC_0a^{n-0}b^0+^nC_1a^{n-1}b^1+...+^nC_na^{n-n}b^n(a+b)
n
=
n
C
0
a
n−0
b
0
+
n
C
1
a
n−1
b
1
+...+
n
C
n
a
n−n
b
n
The value of n of the given expression is 10. It means the middle term is
r=\frac{10}{2}=5r=
2
10
=5
The middle term of the expression is
^nC_5a^{n-5}b^5=^{10}C_5(x)^{10-5}(\frac{-1}{x})^5
n
C
5
a
n−5
b
5
=
10
C
5
(x)
10−5
(
x
−1
)
5
^nC_5a^{n-5}b^5=^{10}C_5(x)^{5}(-\frac{1}{x^5})
n
C
5
a
n−5
b
5
=
10
C
5
(x)
5
(−
x
5
1
)
^nC_5a^{n-5}b^5=-\frac{10!}{5!5!}
n
C
5
a
n−5
b
5
=−
5!5!
10!
^nC_5a^{n-5}b^5=-252
n
C
5
a
n−5
b
5
=−252
Therefore the middle term is -252.
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Question :
Find the middle term in the expansion of (x+1/x)^4 .
Answer :
The middle term in the expansion of (x+1/x^)4 is 6.
Given :
The expression (x+1/x)^4
To find :
The middle term in the expansion of (x+1/x)^4
Solution :
No.of terms = n+1
= 4+1
= 5
Middle term is T3
3 = r+1
=> r =3-1
=> r =2
We know,
T r+1 = nCr x^r y^ (n-r) ---------> (x+y)^n
=> T r+1 = 4Cr (1/x)^(n-r) x^r
=> T 2+1 = 4C2 (1/x)^(4-2) x^2
=> T 3 = 6x^2 . 1/x^2
=> T3 = 6
Hence, the middle term in the expansion of (x+1/x^)4 is 6.
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