Simplifying Series of polynomials with and without fractions

Question

How can I simplify the following two series with $r<1$?

$$(1+r)^{n-1} + (1+r)^{n-2} + \dots + (1+r) + 1$$

and

$$\frac{1}{(1+r)}+\frac{1}{(1+r)^2}+\dots+\frac{1}{(1+r)^{n-1}}+\frac{1}{(1+r)^n}$$

Answer 1

Guide:

Recall for geometric progression:

$$A+AR+AR^2+\ldots+AR^{n-1}=\frac{A(1-R^n)}{1-R}$$

Try to identify the $A$ and $R$ for the two expressions.

Also, remark, we have to assume that $r \ne -1$ for the second expression.

Answer 2

HINT

We have

$$[(1+r)^{n-1} + (1+r)^{n-2} + \dots + (1+r) + 1][1-(1+r)]=1-(1+r)^{n}$$

and

$$\left[\frac{1}{(1+r)}+\frac{1}{(1+r)^2}+\dots+\frac{1}{(1+r)^{n-1}}+\frac{1}{(1+r)^n}\right]=\\=\frac{1}{(1+r)^{n}}[(1+r)^{n-1} + (1+r)^{n-2} + \dots + (1+r) + 1]$$