Prove that terms cyclic in ##a,b,c##, are in ##\text{AP}##

AI Thread Summary
The discussion revolves around proving that the terms a^2(b+c), b^2(c+a), and c^2(a+b) are in arithmetic progression (AP) when a, b, and c are also in AP. The initial approach involved algebraic manipulation of the terms, leading to the conclusion that the differences between consecutive terms are equal, a characteristic of APs. Participants suggested using properties of APs to simplify the proof, such as expressing the terms in a more straightforward manner or leveraging known relationships between the terms. A more efficient method was proposed, involving substitutions that reduce the complexity of the algebra involved. Ultimately, the discussion highlights the importance of recognizing and applying properties of arithmetic progressions to streamline mathematical proofs.
brotherbobby
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Homework Statement
If ##a,b,c## are in ##\text{AP}##, prove that ##\boxed{a^2(b+c), b^2(c+a), c^2(a+b)}## are also in ##\boxed{\text{AP}}##.
Relevant Equations
If terms ##a,b,c## are in ##\text{AP}##, then ##b-a = c-b = d##, where ##d## is the common difference.
Thus ##b = a+d## and ##c = a+2d##.
(This implies that the required expressions in the box above can be reduced to those having only ##a## and ##d##, eliminating ##b## and ##c## from them)
1699528070331.png
Problem statement :
Let me copy and paste the problem as it appears in the text to the right.

Attempt : We have the terms of the ##\text{AP}## as ##a, \;b = a+d, \;c = a+2d##
Let the first term of the required expression be ##t_1 = a^2(b+c) = a^2(2a+3d)=2a^3+3a^2d\dots\quad (1)##
Let the second term of the required expression be ##t_2 = b^2(c+a) = (a+d)^2(a+a+2d) = 2(a+d)^2(a+d) = 2a^3+6a^2d+6ad^2+2d^3\dots\quad (2)##.
Let the third term of the expression be ##t_3= c^2(a+b)=(a+2d)^2(2a+d) = 2a^3+9a^2d+12ad^2+4d^3\dots\quad (3)##
(after some algebra).
Subtracting ##(2) - (1)##, we obtain ##t_2-t_1 = 3a^2d+6ad^2+2d^3## and ##(3)-(2)##, ##t_3-t_2 = 3a^2d+6ad^2+2d^3## *
But this is the property of an ##\text{AP}##, whereby ##t_3-t_2=t_2-t_1##.
Hence the required terms ##\boldsymbol{a^2(b+c), b^2(c+a), c^2(a+b)}## are also in ##\mathbf{\text{AP}}##.

* I can't seem to ##\mathrm{\LaTeX}## this expression. Any clues as to why?

Question (doubt) : I completed the proof, but in a way that is not smart. I wrote out the required terms using ##a## and ##d## and did algebra, but did not use the properties of an AP,
For instance, if ##a_1, a_2, \dots, a_n## are ##n## numbers in AP, then we can show that ##a_1\pm k, a_2\pm k, \dots, a_n\pm k## are also in AP. Likewise, we can also show that ##\frac{pa_1}{q}, \frac{pa_2}{q}, \dots, \frac{pa_n}{q}\quad (p,q\ne0)## are also in AP.
To give you a feel of what I mean, I copy and paste below to the right how the author has answered an easier question of the same type :

1699528197114.png

When I solved this problem before, I again used the method above, viz. express ##c## and ##b## in terms of ##a## and ##d##.

Request : Can someone give a hint of a "smarter" solution to the problem above, whereby I can use the properties of AP and manipulate th variables ##a,b,c## accordingly without resorting to involved algebra?
 
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Another way is to notice that three numbers form an AP if and only if the middle is the average of the other two. So you are given that ##b=\frac{a+c}2## and you need to show that

##b^2(a+c)=\frac{a^2(b+c)+c^2(b+a)}2##

after substitutions this becomes

##\frac{(a+c)^3}4=\frac{a^2(a+3c)+c^2(3a+c)}4##

which can be checked easily. It still needs some algebra though.
 
Re ##\LaTeX ## question: with Reply and toggle BB I see

brotherbobby said:
Code:
[FONT=times new roman]Subtracting ##(2) - (1)##, we obtain ...
 ##t_3-t_2 = [FONT=times new roman]3a^2d+6ad^2+2d^3## 
[FONT=times new roman]*
A blunt copy/paste gives
##t_3-t_2 = 3a^2d+6ad^2+2d^3## So the ##\LaTeX ## is ok.

But a change font encloses the second ## and that breaks the math.

##\ ##
 
It must be slightly easier to let ##a = b -d## and ##c = b +d##. Then the new middle term is ##2b^3##. And it is enough to show that
$$(b+d)^2(2b-d) + (b-d)^2(2b+d) = 4b^3$$Which comes out quite easily.
 
Though, to be overly picky, the author should have stated a,b,c are _ consecutive terms_ in an AP. No reason they can't be the, e.g., 7th, 13th and 59th terms in an AP.
 
I tried to find a reason why this result holds. First, if ##a, b, c## is an AP with common difference ##d##, then ##a+b, a+c, b+c## is an AP with the same difference ##d##. Now, for any ##x, y## there is a unique number ##z## such that ##(a+b)x, (a+c)y, (b+c)z## is an AP. And, in fact$$z = \frac{2(a+c)y - (a+b)x}{b+c}$$The first thing we might try is setting ##x = c, y = b##, in which case ##z = a + \frac{2d^2}{b + c}##, which is not ##a##.

But, if we set ##x = c^2, y = b^2##, then$$z = \frac{2(a+c)b^2 - (a+b)c^2}{b+c} = a^2$$But, it's still not clear why this simplifies to ##a^2##, other then because that's what the algebra tells you!
 
WWGD said:
Though, to be overly picky, the author should have stated a,b,c are _ consecutive terms_ in an AP. No reason they can't be the, e.g., 7th, 13th and 59th terms in an AP.
When we say that three numbers ##a,b,c## are in AP, we don't mean they are a part of an AP. If they are the 7th, 13th and 59th terms of an AP, they wouldn't be in AP themselves.
 
Thank you all for your comments. Your methods work and with less algebra than mine. However, after a search on the internet I have found the type of solution I was looking for. It is smart, but one has to know it in advance I suppose, or else not be able to "invent" it. Since I am a bit late with my response, let me start from the beginning.

1700129680822.png
Problem statement :
Solution :
Please see my attempt in post #1. It was long winded and algebraically involved. I wanted to know if one can use the properties of an AP to solve the problem, keeping the algebra to a minimum.

\begin{equation*}
\begin{split}
a,b,c & \rightarrow \text{in AP} \\
a(ab+bc+ca), b(ab+bc+ca), c(ab+bc+ca) & \rightarrow \text{in AP}\quad\text{(property of AP)}\\
a^2b+abc+a^2c, b^2a+b^2c+abc, abc+c^2b+c^2a & \rightarrow \text{in AP}\\
a^2b+a^2c,b^2a+b^2c, c^2a+c^2b &\rightarrow \text{in AP}\\
\boxed{a^2(b+c),b^2(c+a),c^2(a+b)}&\rightarrow \text{in AP} \quad\color{green}{\huge\checkmark}\\
\end{split}
\end{equation*}
 
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