Demonstration of a Lemma of Archimedes

Lemma to Proposition I of On Conoids and Spheroids

If in an ascending mathematical progression (S) consisting of the magnitudes A₁, A₂, A₃...A_n the common difference will be equal to the least term A₁, then:

the value of (n * A_n) is less than the value 2S_n and more than the value of 2S_n-1.

Demonstration

n * A_n = A is less than:

2S_n = A, but greater than:

2S_n-1 = A