Question

If a b c are in ap then prove that bc/ca+ab,ca/bc+ab, ab/bc+ca that bc/ca+ab, ca/bc+ab, ab/bc+ca are in hp

Answer 1

concept : if $a_1,a_2,a_3,....$ are in ap then,

• $1/a_1,1/a_2,1/a_3....$ are in hp
• $a_1k,a_2k,a_3k,....$ are in ap
• $a_1+k,a_2+k,a_3+k....$ are in ap
• $a_1-k,a_2-k,a_3-k....$ are in ap

by using above concept let's prove it .

a , b , c are in AP.

or, 1/a , 1/b , 1/c are in hp

or, abc/a , abc/b, abc/c are in hp

or, bc , ca , ab are in hp

or, 1/bc , 1/ca , 1/ab are in ap

or, (ab + bc + ca)/bc , (ab + bc + ca)/ca, (ab + bc + ca)/ab are in ap

or, 1 + (ab + ca)/bc, 1 + (ab + bc)/ca, 1 + (bc + ca)/ab are in ap

or, (ab + ca)/bc , (ab + bc)/ca , (bc + ca)/ab are in ap

or, bc/(ab + ca), ca/(ab + bc), ab/(bc + ca) are in hp [ hence proved ]