The A.M. is never below the G.M.
"The A.M. is never below the G.M" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "AM and GM" in Sequences and Series. The line is split into a and b. The radius of the semicircle is the A.M. (a + b)/2. The orange height standing at the split point is the G.M. √(ab). Change a and b and compare the two. For positive a and b, A = (a + b)/2 ≥ G = √(ab), with equality only when a = b. A fixed product means the sum is least when the numbers are equal. JEE Main asked 73 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson AM and GM in Sequences and Series. Change the values and watch what happens.
In JEE Main
42 shifts analysed
73 questions from Sequences and Series across the 42 JEE Main shifts Lumi has analysed (2017–2026).
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.
Questions
3 answers
- What does the "The A.M. is never below the G.M" simulation show?
- The line is split into a and b. The radius of the semicircle is the A.M. (a + b)/2. The orange height standing at the split point is the G.M. √(ab). Change a and b and compare the two.
- Does JEE Main ask questions from Sequences and Series?
- Yes. JEE Main asked 73 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "AM and GM" in Sequences and Series (Class 11 Maths).
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