Question

Find the two positive numbers whose arithmetic mean is 34 and the geometric mean is 16.

Answer 1

See the above photo and please verify...

Answer 2

AnswEr:

Let the two numbers be a and b such that a > b. It is given that AM and GM of a and b are 34 and 16 respectively.

$\qquad \tt \: \frac{a + b}{2} = 34 \: \: \: \: and \: \: \: \: \: \\ \\ \qquad \tt \: \sqrt{ab} = 16 \\ \\ \implies \sf \: a + b = 68 \: \: \: \: and \: \: \: \: ab = 256 \\ \\ \therefore\sf \: {(a - b)}^{2} = {(a + b)}^{2} - 4ab \\ \\ \implies \sf \: {(a - b)}^{2} = {(68)}^{2} - 4 \times 256 = 3600 \\ \\ \implies \sf \: a - b = 60$

Solving a + b = 68 and a - b = 60 simultaneously, we get a = 64 and b = 4.

Hence, the required number are 64 and 4.

ALITER :

Hence, A = 34 and G = 16.

So, the numbers are A + √ A² - G² and A- √A² - G²

i.e. 34 + √34² - 16² and 34 - √34² - 16² or, 64 and 4.