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 layer has 81 boxes and the taop layer has only 1 box, how many boxes are in the display?

285 boxes are in the display

Step-by-step explanation:

Given data

top layer box = 1

last row box = 81

to find out

how many box

solution

we know that every row is a square so that if the bottom layer has 81 squares it mean this is 9² and every row has one lesser box

so that next row will have 8^2 and than 7² and so on till 1²

so we can say that cubes in the rows as that

Sum of all Squares = 9² + 8² + ... + 1²

Sum of Squares positive Consecutive Integers formula are

Sum of Squares of Consecutive Integers = (1/6) (n) (n+1) (2n+1)

here n = 9 so equation will be

Sum of Squares of Consecutive Integers = (1/6) * (9) * (9+1) * (2*9+1)

Sum of Squares of Consecutive Integers = 285

so 285 boxes are in the display