# Exponential formula with decreasing multiplication factor

I am attempting to simply the following calculation where the initial number is $$2.4$$ and add a decrease of $$10$$% from the previous step:

$$1.$$ $$2.4\times 0.9 = 2.16$$

$$2.$$ $$2.4\times\left(0.9 + {0.9}^2\right) = 4.104$$

$$3.$$ $$2.4\times\left(0.9 + {0.9}^2 + {0.9}^3\right) = 5.8536$$

$$4.$$ $$2.4\times\left(0.9 + {0.9}^2 + {0.9}^3 + {0.9}^4\right) = 7.42824$$

$$5.$$ $$2.4\times\left(0.9 + {0.9}^2 + {0.9}^3 + {0.9}^4 + {0.9}^5\right) = 8.845416$$

$$\dots$$ and so on

How could I simply this pattern when the number of adding is X?

• It is very unclear what the question is – Klangen Sep 11 '19 at 8:02
• trying to create a formula where the increasing number of additional power ofs added on is X – Curiosa Sep 11 '19 at 8:10
• This is a simple application of a finite geometric series. – Toby Mak Sep 11 '19 at 8:14

As a hint, consider the geometric series $$a(1+q+q^2+\ldots+q^{n-1}) = a\frac{q^n-1}{q-1}.$$