Geometric mean and problems

The geometric mean (G.M.) of two positive numbers a and b is G = √(ab). It is the number that makes a, G, b a G.P.

• G.M. of 4 and 25: √100 = 10. Check: 10/4 = 2.5 and 25/10 = 2.5, the same ratio.
• Three numbers a, b, c are in G.P. exactly when b² = ac (middle squared = product of the ends), just as they are in A.P. when 2b = a + c.

Inserting several geometric means. To put n numbers between a and b so that the whole list is a G.P., count the terms: there are n + 2, so b is the (n + 2)th term.

• arⁿ⁺¹ = b, so r = (b/a)^(1/(n + 1)).
• Three numbers between 2 and 162: r⁴ = 81, so r = 3 (or –3).
• With r = 3: 6, 18, 54. With r = –3: –6, 18, –54.

A.M. ≥ G.M. For positive a and b, with A = (a + b)/2 and G = √(ab):

• A – G = (a + b – 2√(ab))/2 = (√a – √b)²/2.
• A square is never negative, so A ≥ G, with equality only when a = b.
• For 4 and 25: A = 14.5 and G = 10.

Finding the numbers from A and G. Two positive numbers have A.M. 13 and G.M. 12. Then a + b = 26 and ab = 144.

• Shortcut: (a – b)² = (a + b)² – 4ab = 676 – 576 = 100, so a – b = ±10.
• Add to a + b = 26: the numbers are 18 and 8.
• Or use the quadratic x² – 2Ax + G² = 0: x² – 26x + 144 = 0 has roots 18 and 8.

Growth is a G.P. Anything that is multiplied by the same factor each period grows geometrically.

• ₹2000 at 5% compound interest: after 1 year 2000 × 1.05, after 3 years 2000 × 1.05³ = ₹2315.25.
• 50 bacteria doubling every hour: after n hours there are 50 × 2ⁿ, so after 5 hours 1600.

Be careful about where you start counting. If the term 'after 1 hour' (100) is called a₁, then 'after n hours' is a₁ × 2ⁿ⁻¹ = 100 × 2ⁿ⁻¹, which is the same as 50 × 2ⁿ. Mixing up the starting point shifts every answer by one step.