MHB Ernesto's Question: Proving Harmonic & Arithmetic Progressions

AI Thread Summary
If b+c, a+c, and a+b are in harmonic progression, their reciprocals form an arithmetic progression. By manipulating the equations derived from these relationships, it can be shown that a², b², and c² are also in arithmetic progression. Specifically, the conditions lead to the identity 0=0, confirming the relationship. The proof involves algebraic manipulation of the terms and demonstrating the equality of differences. Thus, the assertion that a², b², and c² are in arithmetic progression holds true under the given conditions.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Prove that if b+c, a+c, a+b are in harmonic progression, then a², b² y c²...?


Prove that if b+c, a+c, a+b are in harmonic progression, then a², b² y c² are in arithmetic progression .

Explain.

Thanks.

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
Hello Ernesto,

If $b+c,\,a+c,\,a+b$ are in harmonic progression, then their reciprocals are in arithmetic progression, and we may state:

(i) $$\frac{1}{a+c}=\frac{1}{b+c}+d$$

Multiply through by $$(a+c)(b+c)$$ to obtain:

$$b+c=a+c+d(a+c)(b+c)$$

Subtract $c$ from both sides and expand the product on the far right:

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

Distribute on the right:

$$b=a+abd+acd+bcd+c^2d$$

Subtract through by $$a+abd+acd+bcd$$:

$$b-a-abd-acd-bcd=c^2d$$

Divide through by $d$ and arrange as:

$$c^2=\frac{b-a-abd-acd-bcd}{d}$$

(ii) $$\frac{1}{a+b}=\frac{1}{b+c}+2d$$

Multiply through by $$(a+b)(b+c)$$ to obtain:

$$b+c=a+b+2d(a+b)(b+c)$$

Subtract $b$ from both sides and expand the product on the far right:

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

Distribute on the right:

$$c=a+2abd+2acd+2b^2d+2bcd$$

Subtract through by $$a+2abd+2acd+2bcd$$:

$$c-a-2abd-2acd-2bcd=2b^2d$$

Divide through by $2d$ and arrange as:

$$b^2=\frac{c-a-2abd-2acd-2bcd}{2d}$$

(iii) $$\frac{1}{a+b}=\frac{1}{a+c}+d$$

Multiply through by $$(a+b)(a+c)$$ to obtain:

$$a+c=a+b+d(a+b)(a+c)$$

Subtract $a$ from both sides and expand the product on the far right:

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

Distribute on the right:

$$c=b+a^2d+acd+abd+bcd$$

Subtract through by $$b+acd+abd+bcd$$:

$$c-b-acd-abd-bcd=a^2d$$

Divide through by $d$ and arrange as:

$$a^2=\frac{c-b-acd-abd-bcd}{d}$$

Now in order for $a^2,\,b^2,\,c^2$ to be in arithmetic progression, we require:

$$b^2-a^2=c^2-b^2$$

Add through by $$a^2+b^2$$:

$$2b^2=a^2+c^2$$

Substitute for $a^2,\,b^2,\,c^2$ the expressions we found above:

$$2\left(\frac{c-a-2abd-2acd-2bcd}{2d} \right)=\frac{c-b-acd-abd-bcd}{d}+\frac{b-a-abd-acd-bcd}{d}$$

Distributing the 2 on the left, and the multiplying through by $d$, we find:

$$c-a-2abd-2acd-2bcd=c-b-acd-abd-bcd+b-a-abd-acd-bcd$$

Add through by $$2abd+2acd+2bcd$$:

$$c-a=c-b+b-a$$

Collect like terms:

$$c-a=c-a$$

Subtract through by $$c-a$$:

$$0=0$$

This is an identity, which proves that given $b+c,\,a+c,\,a+b$ are in harmonic progression, then $a^2,\,b^2,\,c^2$ must be in arithmetic progression.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top