SOLUTION: if a,b,c are in HP and ab + bc + ca = 15 then ca =

Algebra ->  Sequences-and-series -> SOLUTION: if a,b,c are in HP and ab + bc + ca = 15 then ca =       Log On


   

Question 844224: if a,b,c are in HP and ab + bc + ca = 15 then ca =
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
if a,b,c are in HP and ab + bc + ca = 15 then ca =
Since a,b,c are in harmonic progression, their 
reciprocals 1%2Fa,1%2Fb,1%2Fc form an arithmetic sequence:

Let d be the common difference.  Then

1%2Fa%2Bd%22%22%2B%22%221%2Fb
1%2Fb%2Bd%22%22%2B%22%221%2Fc

Solve each for d

d%22%22=%22%221%2Fb-1%2Fa
d%22%22=%22%221%2Fc-1%2Fb

Since both equal d, they are equal to each other:

1%2Fb-1%2Fa%22%22=%22%221%2Fc-1%2Fb

Multiply through by LCD of abc

ac - bc = ab - ac

2ac = ab + bc

And since we are given that

ab + bc + ca = 15

We can substitute 2ac for ab + bc and get

2ac + ca = 15

And since ac is the same as ca

2ca + ca = 15

3ca = 15

ca = 5

Edwin