Proving Binomial Theorem with Greatest Term and Coefficient Relationship

AI Thread Summary
To prove that the greatest term in the expansion of (1+x)2n is also the greatest coefficient, it is established that x must lie between n/n+1 and n+1/n. The greatest coefficient occurs at the nth term due to the even nature of 2n. By analyzing the coefficients of the (n-1)th and (n+1)th terms, it is shown that they are equal. Plugging in the specified boundaries for x and performing algebra reveals that at one boundary the (n-1)th and nth terms are equal, while at the other boundary the nth and (n+1)th terms are equal. This confirms the relationship between the greatest term and coefficient in the binomial expansion.
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Homework Statement



Show that if the greatest term in the expansion of (1+x)2n is also the greatest coefficient, then x lies between n/n+1 and n+1/n.

Homework Equations



No idea.

The Attempt at a Solution



Don't know where to start.
 
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2n is obviously an even number, so your greatest coefficient occurs at the nth term. Use the binomial expansion to also find your (n-1)th and (n+1)th terms (their coefficients will be equal), plug in your given boundaries for x, and do a little algebra to show that at one of the boundaries the (n-1)th and nth terms are equal and at the other boundary the nth and (n+1)th terms are equal. Hopefully this leads you in the right direction.
 
Thank You, it is clear now.
 

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