BTGmoderatorDC wrote: ↑Tue Mar 17, 2020 6:51 pm
For a display, identical cubic boxes are stacked in square layers. Each layer consists of cubic boxes arranged in rows that form a square, and each layer has 1 fewer row and 1 fewer box in each remaining row than the layer directly below it. If the bottom of the layer has 81 boxes and the top of the layer has only 1 box, how many boxes are in display?
A. 236
B. 260
C. 269
D. 276
E. 285
OA
E
Source: GMAT Prep
Since it is given that the no. of cubes in the bottom row is 81, and they are arranged in a square, there are √81 = 9 rows.
Let's find out the no. of cubes in each layer.
• No. of cubes in the bottom/9th row = 81 = 9^2;
• No. of cubes in the 8th row = (9 – 1)*(9 – 1) = 8^2 = 64;
• No. of cubes in the 7th row = 7^2 = 49;
• No. of cubes in the 6th row = 6^2 = 36;
• No. of cubes in the 5th row = 5^2 = 25;
• No. of cubes in the 4th row = 4^2 = 16;
• No. of cubes in the 3rd row = 3^2 = 9;
• No. of cubes in the 2nd row = 2^2 = 4;
• No. of cubes in the 1st row = 1^2 = 1
Sum of all cubes = 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 285
Alternatively, we apply the formula for the sum of squares of the first n positive integers.
SUM = n(n+1)(2n + 1)/6; where n = no. of positive integers, here n = 9
Thus, SUM = 9*(9+1)(2*9 + 1)/6 = 285
The correct answer:
E
Hope this helps!
-Jay
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