Inserting k Arithmetic Means between a and b: The k means form an AP with a and b: a, A_1, A_2, ..., A_k, b Total terms = k+2, so d = (b-a)/(k+1) A_i = a + i*d = a + i(b-a)/(k+1) Sum of k AMs = k(a+b)/2

Inserting k Geometric Means between a and b: The k means form a GP with a and b: a, G_1, G_2, ..., G_k, b Total terms = k+2, so r = (b/a)^{1/(k+1)} G_i = a * r^i Product of k GMs = (ab)^{k/2}

Key result: The sum of n AMs between a and b = n times the single AM = n(a+b)/2. Similarly, the product of n GMs = (GM)^n = (ab)^{n/2}.