# Maximum term of (a + b) ^ n

I would like a demonstration of the fact below.

Being given real numbers a and b (nonzero) and a positive integer n, the order p, that occupies the maximum term (in absolute value) of the development of power (a+b)^ n, according to decreasing powers of a is given by:

p = 1 + integer part of [|b|(n+1)/(|a|+|b|)]

When n is integer, there are maximum two terms: those of order p and p-1.

|a| and |b| are the modules of the numbers a and b, respectively.

Paulo Argolo

Rio de Janeiro, Brazil

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Let $f(k)=C_n^k |a|^{n-k} |b|^k$.

Then $f(k+1)/f(k)=\frac{|b|}{|a|} \frac{n-k}{k+1} > 1$ iff $k<\frac{|b|(n+1)}{(|b|+|a|)}-1$, so $f(k)$ is increasing until $k=I\left(\frac{|b|(n+1)}{(|b|+|a|)}\right)$ where $I$ denotes the integral part.

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Unfortunately, I could not understand your resolution. The notation used is very confusing to me. I have difficulty with the symbols used. Even so, many thanks – Paulo Argolo Nov 5 '10 at 0:09
Maybe the notation $C^k_n$ is confusing you, it is also written $\binom{k}{n}$ and equal to $\frac{n!}{k!(n-k)!}$. To better understand the proof you can also do the case $a=b$, where only the binomial coefficients "count". – Plop Nov 5 '10 at 1:21
$\binom{n}{k}$, not $\binom{k}{n}$. – Hans Lundmark Nov 5 '10 at 7:50
I needed to correct the statement. There were errors. – Paulo Argolo Nov 5 '10 at 16:25

Every positive integer is of the form 3k-2,3k-1, or 3k.

1. If n is of the form 3k-2, then ak has the maximum value
2. If n is of the form 3k-1, then ak = a[k+1] have the maximum value.
3. If n is of the form 3k, then a[k+1] has the maximum value.

It's not hard to prove each of those. Write out the terms with the factorials and powers of 2 for positive j<=k and prove: a[k-j] < ak {and/or a[k+1]} and ak {and/or a[k+1]} > a[k+1+j]

Edwin McCravy

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Welcome to MSE! I am sorry to tell you that you got off to kind of a bad start. Mathematically, your formatting is nonstandard, and the end result is quite different than the other answer which is well-justified. Also, you should not sign your name at the bottom of your posts; all of your posts come with a signature in the form of a link to your user page. – Eric Stucky Jul 1 '14 at 2:24