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 = $\frac{b-a}{(k+1)}$ $A_i$ = a + i*d = a + i$\frac{b-a}{(k+1)}$ Sum of k AMs = k$\frac{a+b}{2}$

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 = $\frac{b}{a}$^{$\frac{1}{k+1}$} $G_i$ = 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$\frac{a+b}{2}$. Similarly, the product of n GMs = (GM)^n = (ab)^{n/2}.