Show that three numbers form an arithmetic progression

Question

The numbers $a,b$ and $c$ form an arithmetic progression. Show that the numbers $a^2+ab+b^2,a^2+ac+c^2,b^2+bc+c^2$ also form an arithmetic progression.

We have that $2b=a+c$ (we know that a sequence is an arithmetic progression iff $a_n=\dfrac{a_{n-1}+a_{n+1}}{2}\text{ } \forall \text{ }n\ge2$). I am stuck here and I would be very grateful if you could give me a hint.

Answer 1

$(a^2+ac+c^2)-(a^2+ab+b^2)=a(c-b)+c^2-b^2=(c-b)(a+b+c)\\(b^2+bc+c^2)-(a^2+ac+c^2)=b^2-a^2+(b-a)c=(b-a)(a+b+c)$

Are they equal?

Answer 2

More hint:

\begin{align}&\quad(a^2+ab+b^2) + (b^2+bc + c^2) \\&= a^2+c^2 +2b^2 + b(a+c) \\&= a^2 + c^2 + 2b^2 + b(2b) \\&= a^2 + c^2 + 4b^2\\&=a^2+c^2+(a+c)^2\end{align}

In the worst case you can just substitute $b = \frac {a+c}2$.

Answer 3

Since $2b=a+c$

$$a^2+ab+b^2+b^2+bc+c^2=a^2+b(a+c)+2b^2+c^2=a^2+\frac{(a+c)^2}{2}+2\times \frac{(a+c)^2}{4}+c^2=a^2+(a+c)^2+c^2=2(a^2+ac+c^2)$$

Answer 4

Well, If $a,b,c$ form an arithmetic progression then there is a $d$ so that $a=b-d$ and $c=b+d$.

And to show that $a^2+ab+b^2,a^2+ac+c^2,b^2+bc+c^2$ form an arithmetic progression we have to show there is a $k$ so that

$k = (a^2+ac+c^2) - (a^2+ab+b^2)$ and $k= (b^2+bc+c^2) -(a^2+ac+c^2)$ or in other words that

$(a^2+ac+c^2) - (a^2+ab+b^2)= (b^2+bc+c^2) -(a^2+ac+c^2)$. So if we replace $a=b-d$ and $c = b+d$ we must show

$((b-d)^2+(b-d)(b+d)+(b+d)^2) - ((b-d)^2+(b-d)b+b^2)= (b^2+b(b+d)+(b+d)^2) -((b-d)^2+(b-d)(b+d)+(b+d)^2)$

Can we?

$((b-d)^2+(b-d)(b+d)+(b+d)^2) - ((b-d)^2+(b-d)b+b^2)=$

$((b-d)(b+d)+(b+d)^2) - ((b-d)b+b^2)=$

$((b-d)d+(b+d)^2) - (b^2)=$

$(b-d)d + 2bd+d^2=$

$3bd $.

And

$(b^2+b(b+d)+(b+d)^2) -((b-d)^2+(b-d)(b+d)+(b+d)^2)=$

$(b^2+b(b+d)) -((b-d)^2+(b-d)(b+d))=$

$(b^2) -((b-d)^2-d(b+d))=$

$(b^2) -(b^2-2bd-bd)=$

$3bd$.

Yep, they are equal all right.

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To take your idea of $a_n=\dfrac{a_{n-1}+a_{n+1}}{2}\text{ } \forall \text{ }n\ge2$ into account:

We know $b = \frac{a+c}2$ and we need to prove

$a^2+ac+c^2= \frac {(a^2+ab+b^2)+(b^2+bc+c^2)}2$ given that $b = \frac {a+c}2$

So

$\frac {(a^2+ab+b^2)+(b^2+bc+c^2)}2=$

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

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

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

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

$\frac {a^2 + c^2 + a^2 + 2ac + c^2}2 =$

$\frac {2a^2 + 2ac + 2c^2}2 =$

$a^2 + ac + c^2$.

Answer 5

Expressions like $ \ a^2+ab+b^2 \ , \ a^2+ac+c^2 \ , \ b^2+bc+c^2 \ \ $ make me think of "differences of two cubes", so let's see what happens. If we call the progression $ \ a \ = \ b - d \ , \ b \ , \ c \ = \ b + d \ \ , $ then we may write $$ \ a^2 \ + \ ab \ + \ b^2 \ \ = \ \ \frac{b^3 \ - \ a^3}{b - a} \ \ = \ \ \frac{b^3 \ - \ (b - d)^3}{ d} \ \ = \ \ \frac{3·b^2d \ - \ 3·bd^2 \ + \ d^3}{d} \ \ , $$ $$ \ a^2 \ + \ ac \ + \ c^2 \ \ = \ \ \frac{c^3 \ - \ a^3}{c - a} \ \ = \ \ \frac{(b + d)^3 \ - \ (b - d)^3}{2d} \ \ = \ \ \frac{6·b^2d \ + \ 2·d^3}{2d} \ \ , $$ $$ \ b^2 \ + \ bc \ + \ c^2 \ \ = \ \ \frac{c^3 \ - \ b^3}{c - b} \ \ = \ \ \frac{(b + d)^3 \ - \ b^3}{d} \ \ = \ \ \frac{3·b^2d \ + \ 3·bd^2 \ + \ d^3}{d} \ \ . $$

The expressions in this sequence simplify to $$ 3·b^2 \ - \ 3·bd \ + \ d^2 \ \ , \ \ 3·b^2 \ + \ d^2 \ \ , \ \ 3·b^2 \ + \ 3·bd \ + \ d^2 \ \ , $$ which differ sequentially by a constant amount $ \ 3·bd \ \ . $