Simulation · Maths · Class 11

The A.M. is never below the G.M.

From the lesson AM and GM in Sequences and Series. Change the values and watch what happens.

The idea behind it

NCERT §8.5

For positive numbers a and b, the arithmetic mean is A = (a + b)/2 and the geometric mean is G = √(ab).
A − G = (a + b − 2√(ab))/2 = (√a − √b)²/2, which is never negative. So A ≥ G for all positive a and b.
Equality A = G holds only when √a = √b, that is when a = b. For 2 and 8, A = 5 and G = 4.
The two means fix the numbers. If A = 10 and G = 8 then a + b = 20 and ab = 64, so (a − b)² = 400 − 256 = 144, a − b = ±12, and the numbers are 4 and 16.
Another pair: A = 13 and G = 12 give a + b = 26, ab = 144, (a − b)² = 676 − 576 = 100, so the numbers are 8 and 18.
Numbers with A and G known are the roots of x² − 2Ax + G² = 0. For A = 5 and G = 4 that is x² − 10x + 16 = 0, with roots 2 and 8.
A ≥ G gives quick bounds: for positive x, x + 1/x ≥ 2; two positive numbers with product 36 have sum at least 12; a rectangle with perimeter 40 has area at most 10 × 10 = 100.

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