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If 1/a+1/a-b+1/c+1/c-b =0 and a+c-b not equal to 0 then prove a,b,c are in hp

Harshidaa S Nair , 7 Years ago
Grade 11
anser 1 Answers
Ritesh Khatri

Last Activity: 7 Years ago

After rearranging we get
 
\Rightarrow \frac{1}{a}+\frac{1}{c-b}+\frac{1}{a-b}+\frac{1}{c}=0
After taking LCM of first two and last to terms we get
 
\Rightarrow \frac{c-b+a}{a(c-b)}+\frac{c+a-b}{c(a-b)}=0
since a+c-b is not equal to 0
 
therefore      \frac{1}{a(c-b)}+\frac{1}{c(a-b)}=0
 
Taking LCM
 
\Rightarrow \frac{ac-bc+ac-ab}{ac(c-b)(a-b)}=0
 
Therefore numerator is equal to zero
 
\Rightarrow \ ac-bc+ac-ab = 0
 
\Rightarrow \ 2ac = bc+ab
 
\Rightarrow \ \frac{2}{b} = \frac{1}{a}+\frac{1}{c}
Which is necessory condition to proove that a, b and c are in HP
 

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