Originally Posted by
SlipEternal I will help you out by providing some notation. The arithmetic progression can be represented by the terms $\displaystyle a_0, \ldots, a_n$ and the geometric progression can be represented by the terms $\displaystyle g_0, \ldots, g_n$. For a progression to be arithmetic, there must be some real number $\displaystyle d$ such that $\displaystyle a_k = a_0 + kd$. For a progression to be geometric, there must be some real number $\displaystyle r$ with $\displaystyle g_k = g_0r^k$. Next, let's figure out the arithmetic and geometric sums:
$\displaystyle A = \sum_{k=0}^n a_k = \sum_{k=0}^n (a_0 + kd) = a_0\sum_{k=0}^n 1 + d\sum_{k=0}^n k$
$\displaystyle G = \sum_{k=0}^n g_k = \sum_{k=0}^n g_0r^k = g_0\sum_{k=0}^n r^k$