SOLUTION: If a,b,c are in Harmonic Progression,prove that 1/a + 1/(b+c), 1/b + 1(c+a), 1/c + 1(a+b) are also in Harmonic Progression

Algebra ->  Sequences-and-series -> SOLUTION: If a,b,c are in Harmonic Progression,prove that 1/a + 1/(b+c), 1/b + 1(c+a), 1/c + 1(a+b) are also in Harmonic Progression       Log On


   

Question 1210195: If a,b,c are in Harmonic Progression,prove that 1/a + 1/(b+c), 1/b + 1(c+a), 1/c + 1(a+b) are also in Harmonic Progression
Answer by mccravyedwin(407) About Me  (Show Source):
You can put this solution on YOUR website!

If a,b,c are in Harmonic Progression,

That means that 1/a, 1/b, 1/c are in arithmetic progression, i.e.

1%2Fb-1%2Fa=1%2Fc-1%2Fb

ac-bc=ab-ac
2ac=ab%2Bbc
2ac=b%28a%2Bc%29
b=%282ac%29%2F%28a%2Bc%29

We are to show that

(1)   1%5E%22%22%2F%281%2Fb+%2B+1%28c%2Ba%29%29-+1%5E%22%22%2F%281%2Fa+%2B+1%2F%28b%2Bc%29%29%22%22=%22%221%5E%22%22%2F%281%2Fc+%2B+1%28a%2Bb%29+%29-1%5E%22%22%2F%281%2Fb+%2B+1%28c%2Ba%29%29


All terms are of the form 1%5E%22%22%2F%281%2Fx+%2B+1%2F%28y%2Bz%29%29

Multiply top and bottom by x(y+z)

%28x%28y%2Bz%29%29%2F%28y%2Bz%2Bx%29%29

So (1) becomes

(2)   %28b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29-%28a%28b%2Bc%29%29%2F%28b%2Bc%2Ba%29%22%22=%22%22%28c%28a%2Bb%29%29%2F%28a%2Bb%2Bc%29-%28b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29 

(2) will be true if and only if

%282b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29-%28a%28b%2Bc%29%29%2F%28b%2Bc%2Ba%29%22%22=%22%22%28c%28a%2Bb%29%29%2F%28a%2Bb%2Bc%29 

which will be true if and only if

2b%28c%2Ba%29-a%28b%2Bc%29%22%22=%22%22c%28a%2Bb%29

which will be true if and only if 

2bc%2B2ab-ab-ac%22%22=%22%22ac%2Bbc

which will be true if and only if

bc%2Bab-2ac%22%22=%22%220

Substitute b=%282ac%29%2F%28a%2Bc%29

%28%282ac%29%2F%28a%2Bc%29%29c%2Ba%28%282ac%29%2F%28a%2Bc%29%29-2ac%22%22=%22%220

which will be true if and only if

2ac%5E2%2B2a%5E2c-2ac%28a%2Bc%29+=+0

Which will be true if and only if 

c%2Ba+-+%28a+%2B+c%29+=+0

And since this is true, the problem is proved.

Edwin