swerve wrote:A table of numbers has n rows and n columns, where n is an odd integer greater than 4. Each entry in the table is subject to the following rules:

(I) Any entry is the negative of the entry directly above it, if there is an entry directly above it.
(II) Any entry is double the entry directly to its left, if there is an entry directly to its left.

If the entry in the upper left corner (first row and first column) is 1, what is the sum of all the entries in the table?

A. 0
B. n^2 - 1
C. n^2 + 1
D. 2^n - 1
E. 2^n + 1

The OA is D

Source: Manhattan Prep

Say n = 5

As per the two conditions, we have 5*5 grid as following.

-----------------------------
| 1 | 2 | 4 | 8 | 16 |
------------------------------
| -1 | -2 | -4 | -8 | -16 |
------------------------------
| 1 | 2 | 4 | 8 | 16 |
------------------------------
| -1 | -2 | -4 | -8 | -16 |
-----------------------------
| 1 | 2 | 4 | 8 | 16 |
------------------------------
We see that each entry in row 1 has a corresponding negative entry in row 2, making the sum of all the entries in row 1 and row 2 equals to 0; same goes with row 3 and row 4.

So, only the sum of entries of row 5 matters, Sum of the entries in row 5 = 1 + 2 + 4 + 8 + 16 = 31

Since the options are given in terms of the variable n, we can plug-in n = 5 to get the correct answer.

At n = 5, we have option D = 2^n - 1 = 2*5^2 - 1 = 31 (correct answer).

The correct answer: D

Hope this helps!

-Jay
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