if a,b,c are in AP
Then prove that a (1/b + 1/c ) , b (1/c + 1/a ), c (1/a + 1/b ) are in AP

Dear student,

Please find below the solution to the asked query:

Since a, b and c are in AP. This gives    b-a=c-b   .....1Consider the following terms.   a1b+1c, b1c+1a , c1a+1bFind the difference of cosecutive terms.   b1c+1a-a1b+1c=bc+ba-ab-ac                                                =ab2+cb2-a2c-a2babc                                                =ab2-a2b+cb2-a2cabc                                                =abb-a+cb2-a2abc                                                =abb-a+cb-ab+aabc                                                =b-aab+cb+aabc                                                =b-aab+bc+caabcand     c1a+1b-b1c+1a=ca+cb-bc+ba                                                =bc2+ac2-ab2-cb2abc                                                =bc2-cb2+ac2-ab2abc                                                 =bcc-b+ac2-b2abc                                                =bcc-b+ac-bc+babc                                                =c-bbc+ac+babc                                                =c-bab+bc+caabcNote that both are difference have same terms except b-a  and c-bBut this is already equal using 1This shows that difference between the consecutive terms is same.This proves that a1b+1c, b1c+1a , c1a+1b are in AP

Hope this information will clear your doubts about the topic.                      

If you have any more doubts, just ask here on the forum and our expert will try to help you out as soon as possible.                       

Regards

 

  • -4
What are you looking for?