SOLUTION: If b is the harmonic mean between a and c,then prove that:1/(b-a) +1/(b-c)=1/a+1/c

Algebra ->  Sequences-and-series -> SOLUTION: If b is the harmonic mean between a and c,then prove that:1/(b-a) +1/(b-c)=1/a+1/c      Log On


   

Question 1083841: If b is the harmonic mean between a and c,then prove that:1/(b-a) +1/(b-c)=1/a+1/c
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
We find the arithmetic mean of 1%2Fa and 1%2Fc

Add them 1%2Fa%2B1%2Fc=%28a%2Bc%29%2F%28ac%29

Divide by 2:  %28a%2Bc%29%2F%282ac%29

Its reciprocal %282ac%29%2F%28a%2Bc%29 is the harmonic mean
between a and c, so

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

Substitute for b in

1%2F%28b-a%29+%2B1%2F%28b-c%29

1%2F%28%28%282ac%29%2F%28a%2Bc%29%29-a%29+%2B1%2F%28%28%282ac%29%2F%28a%2Bc%29%29-c%29

Multiply numerators and denominators by (a+c)

%28a%2Bc%29%2F%282ac-a%28a%2Bc%29%29+%2B%28a%2Bc%29%2F%282ac-c%28a%2Bc%29%29

%28a%2Bc%29%2F%282ac-a%5E2-ac%29+%2B%28a%2Bc%29%2F%282ac-ac-c%5E2%29%29

%28a%2Bc%29%2F%28ac-a%5E2%29+%2B%28a%2Bc%29%2F%28ac-c%5E2%29%29

%28a%2Bc%29%2F%28a%28c-a%29%29+%2B%28a%2Bc%29%2F%28c%28a-c%29%29%29

%28a%2Bc%29%2F%28a%28c-a%29%29+%2B%28a%2Bc%29%2F%28c%28-c%2Ba%29%29%29

%28a%2Bc%29%2F%28a%28c-a%29%29+%2B%28a%2Bc%29%2F%28-c%28c-a%29%29%29

%28a%2Bc%29%2F%28a%28c-a%29%29+-%28a%2Bc%29%2F%28c%28c-a%29%29%29

Factor out %28a%2Bc%29%2F%28c-a%29

%28%28a%2Bc%29%2F%28c-a%29%29%281%2Fa-1%2Fc%29%29%29

%28%28a%2Bc%29%2F%28c-a%29%29%28%28c-a%29%2F%28ac%29%29%29%29

%28%28a%2Bc%29%2F%28cross%28c-a%29%29%29%28%28cross%28c-a%29%29%2F%28ac%29%29%29%29

%28%28a%2Bc%29%2F%28ac%29%29%29%29

a%2F%28ac%29%2Bc%2F%28ac%29

cross%28a%29%2F%28cross%28a%29c%29%2Bcross%28c%29%2F%28a%2Across%28c%29%29

1%2Fc%2B1%2Fa

1%2Fa%2B1%2Fc

Edwin