Tutoring math, geometric sequences are commonly encountered.

You’ll find an introduction to geometric sequences in my post here.

Example: Let’s imagine a geometric sequence with ratio r greater than one and start term M/r. It runs

M/r, M, Mr….

Can we be sure that the successive differences always increase?

Solution:

We need to show that Mr-M > M-M/r.

We get a common denominator for M-M/r:

M-M/r=(Mr-M)/r

Since r>1, (Mr-M) > (Mr-M)/r. Since (Mr-M)/r = M-M/r, we see that, indeed, Mr-M > M-M/r. When r>1, the differences between successive terms in a geometric sequence must continually increase.

Source:

Travers, Kenneth et al. Using Advanced Algebra. Toronto: Doubleday, 1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.