If the geometric mean of a and b is 10 and the harmonic mean of a and b is 8, find the value of the following without determining the values of a and b.
1/a + 1/b
Hello, Mr_Green!
Are you sure of the wording of the problem?
. . It seems to be much too simple.
If the geometric mean of a and b is 10 . ← not needed
and the harmonic mean of a and b is 8,
find the value of (1/a + 1/b) without determining the values of a and b.
. . . . . . . . . . . . . 2
We are given: . -------- . = . 8
. . . . . . . . . . . .1 . . 1
. . . . . . . . . . . .-- + --
. . . . . . . . . . . .a . . b
. . . . . .1 . . 1 . . . . 1
Then: . -- + -- . = . --
. . . . . .a . . b . . . . 4
Hello again, Mr_Green!
We are given: .√(ab) = 10 . → . ab = 100 . → . 2/ab = 1/50 .[1]If the geometric mean of a and b is 10 and the harmonic mean of a and b is 8,
find the value of: .1/aČ + 1/bČ
I have already shown that: .1/a + 1/b .= .1/4
Square both sides: .1/aČ + 2/ab + 1/bČ .= .1/16
. . . . . . . . . . . . . . . . . . . . ↓
Substitute [1]: . . . 1/aČ + 1/50 +1/bČ .= .1/16
Therefore: .1/aČ + 1/bČ .= .1/16 - 1/50 .= .17/400