Question

If b is the harmonic mean between a and c, prove that
\[\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}\]

Answer 1

hmn wait lemme open my book .. wait

Answer 2

If am not wrong this must be the meaning:-
Harmonic Mean of a set of numbers is the number of items divided by the sum of the reciprocals of the numbers. Hence, the Harmonic Mean of a set of n numbers

Answer 3

i am confused by your definition.
i know that H.M. is the reciprocal of A.M.

Answer 4

SO you can proceed further @shubham.bagrecha as Harmonic Mean of a set of numbers is the number of items divided by the sum of the reciprocals of the numbers.

Answer 5

So hence proven

Answer 6

how ??

Answer 7

basic formula :
\[\large{\textbf{Harmonic Mean}=\frac{2ab}{a+b}}\]

Answer 8

lets start with LHS
\[\frac{1}{b-a}+\frac{1}{b-c}=\frac{2b-(a+c)}{b ^{2}+ac-b(a+b)}\]
now since they are in HP,
\[a+c=\frac{2ac}{b}\]
substituting this,we get
\[\frac{2b-\frac{2ac}{b}}{b ^{2}+ac-2ac}=\frac{2}{b}\]
and accordind to defination of HP,u get that as 1/c+1/a.....

Answer 9

the better formula for this case will be :
let the numbers be a, b and c

so we have :
\[\large{\frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}}\]
\[\large{\frac{2}{b}=\frac{a+c}{ac}}\]
\[\large{b=\frac{2ac}{a+c}}\]

Answer 10

^^

Answer 11

good work @hartnn