If 4 and b have the same arithmetic as well as geometric means, what is the value of b?  

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

Given that the numbers 4 and b have the same geometric and arithmetic means.

Then we know that:

Arithmetic mean = ( 4 + b ) / 2........(1)

Geometric mean =  sqrt(4*b)............(2)

Given that (1) = (2).

==> ( 4 + b) /2 = sqrt(4b)

We will mutiply by 2.

==> (4+ b) = 2sqrt(4b)

Now we will square both sides.

==> (4+ b)^2 = 4*4b

==> 16 + 8b + b^2 = 16b

==> b^2 - 8b + 16 = 0

==> ( b -4)^2 = 0

==> b = 4

To check:

Arithmetic mean = ( 4+ 4) / 2 = 8/2 = 4

Geometric mean = sqrt(4*4) = sqrt16 = 4

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The arithmetic mean of two number a and b is given by (a + b)/2. The geometric mean of a and b is given by sqrt (a*b).

As the arithmetic mean and geometric mean of 4 and b is equal, we have:

(4 + b)/2 = sqrt 4b

=> 4 + b = 2 sqrt 4b

=> (4 + b) ^2 = 4*4b

=> 16 + b^2 + 8b = 16b

=> 16 + b^2 – 8b = 0

=> (b – 4) ^2 = 0

=> b – 4 = 0

=> b = 4

Therefore the value of b is 4.

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