Mock AMC 8 #1 - Problem 17

Munch has a 3\times3 grid with the following rules:

1. The first and second rows from the top form an arithmetic series
2. Every column forms a geometric series.
3. The main diagonal from top left to bottom right is both an arithmetic and geometric series.

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant (the common difference) to the previous term. For example, 4, 7, 10, 13, 16 is an arithmetic sequence with 5 terms and a common difference of 3.

A geometric sequence is a sequence of numbers such that the ratio between any two consecutive terms is constant. This constant is called the common ratio of the sequence. For example, 1, 2, 4, 8 is a geometric sequence with 4 terms and a common ratio of 2.

Several of the cells are already filled, as shown. What is the sum of the values of every cell?

(\textbf{A})\;32\quad(\textbf{B})\;35\quad(\textbf{C})\;33\quad(\textbf{D})\;34\quad(\textbf{E})\;38