It seems exponential progression and geometric progression both mean the same thing in common language.

But geometric progression is getting to the next term by multiplication of a constant "common ratio".

For example: 2, 4, 8, 16, 32... (a_{1}= 2, common ratior = 2)

Of course the partial sums increase at a non-linear rate.

Yes this might be non-linear, but for me, it doesn't really fit "exponential" that well because it works by multiplication.

What if we had progression that worked up the next level of operations, exponentiation, as in powers?

Here, a term is taken to a constant "common power" to produce the next term.

Like this: 2, 4, 16, 256, 65536... (a_{1}= 2, common powerp= 2)

So could we apply this to the real world?