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Consider the following statements
I. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is equal to the middle term of $\left(x+\frac{1}{x}\right)^{8}$.
II. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is less than the coefficient of the fifth term in the expansion of $(1+x)^{7}$. Which of the above statements is/ are correct?
Options:
I. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is equal to the middle term of $\left(x+\frac{1}{x}\right)^{8}$.
II. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is less than the coefficient of the fifth term in the expansion of $(1+x)^{7}$. Which of the above statements is/ are correct?
Solution:
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Verified Answer
The correct answer is:
I only
Statement I : Given expansion is $(1+x)^{8}$ Since, $n=8$ is even
$\therefore\left(\frac{n}{2}+1\right)$ th term is the middle term.
ie- $\left(\frac{8}{4}+1\right)^{\text {th }}=5^{\text {th }}$ term $=$ middle term
Now, $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4} 1^{4}={ }^{8} C_{4} x^{4}$
Coeff of $5^{\text {th }}$ term (middle term) $={ }^{8} C_{4}$
Now, consider the expansion $\left(x+\frac{1}{x}\right)^{8}$
It's middle term $=5^{\text {th }}$ term
and $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4}\left(\frac{1}{x}\right)^{4}={ }^{8} C_{4}$
Hence, statement I is correct.
Statement-II : Coeff. of middle term in $(x+1)^{8}$ is
${ }^{8} C_{4}=\frac{8 !}{4 ! 4 !}=70$
Coeff of $5^{\text {th }}$ term in $(1+x)^{7}={ }^{7} C_{4}=\frac{7 !}{4 ! 3 !}=35$
Hence, statement II is incorrect.
$\therefore\left(\frac{n}{2}+1\right)$ th term is the middle term.
ie- $\left(\frac{8}{4}+1\right)^{\text {th }}=5^{\text {th }}$ term $=$ middle term
Now, $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4} 1^{4}={ }^{8} C_{4} x^{4}$
Coeff of $5^{\text {th }}$ term (middle term) $={ }^{8} C_{4}$
Now, consider the expansion $\left(x+\frac{1}{x}\right)^{8}$
It's middle term $=5^{\text {th }}$ term
and $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4}\left(\frac{1}{x}\right)^{4}={ }^{8} C_{4}$
Hence, statement I is correct.
Statement-II : Coeff. of middle term in $(x+1)^{8}$ is
${ }^{8} C_{4}=\frac{8 !}{4 ! 4 !}=70$
Coeff of $5^{\text {th }}$ term in $(1+x)^{7}={ }^{7} C_{4}=\frac{7 !}{4 ! 3 !}=35$
Hence, statement II is incorrect.
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