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Lets say n = 5

Constructed the table below:

Only numbers present in yellow shaded region remain; rest all add up to 0

Final addition = 16 + 8 + 4 + 2 + 1 = 31

\(31 = 2^5 - 1\)

\(Answer = 2^n - 1 = D\)
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sq.jpg [ 18.99 KiB | Viewed 9266 times ]

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I created a table using n=5 (smallest case). Since we have negative/positive changes between rows, the sums will cancel each other out on the even rows (eg. rows 1 + 2 cancel, 3 + 4 cancel, etc). In the case of n=5, all we are left with is row #5. The entries in row 5 (left to right), n=5 is 1, 2, 4, 8, 16. Summing this gives you 31. Plugging n=5 into the answer choices, D gives you the correct answer.

Answer is D.
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Answer = D

Refer diagram below:(5*5 matrix)

Attachment:
mat.png
mat.png [ 2.59 KiB | Viewed 9337 times ]

Only numbers in yellow remain

Sum = 1+2+4+8+16 = 31

\(2^5 - 1 = 31\)

\(Answer = 2^n - 1\)
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2^n means 2+2+(2+2)+(2+2+2+2)+(2+2+2+2+2+2+2+2)+...
since we double each entry, we go like this: 1+1*2+1*2*2+1*2*2*2 which is basically the same as above but with 1 in the beginning instead of 2. so if we calculate 2^n and remove the difference between 1 and 2 => -1,
we get: 2^n-1 (D)
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Given that n is odd.

The first entry is 1. The remaining entries in the 1st row will be 1,2,4,8,16....

The entry in the 2nd row will be the negative of the entries in the 1st row.

Similarly, the entries in the 3rd row will negate those in the 2nd row.

Therefore, if there were even number of rows, the entries would be equally negated. Then the sum would have been 0.

Since there will be one row left (n is odd), the sum will be equal to the entry in the first row.

It will be equal to the sum of the powers of 2, which is given by (2^n) - 1.
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