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