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(B) The pyramid has $18$ layers,with the top layer having $13$ balls on each side. Since it is a square-based pyramid,the number of balls in the $k$-t

Vedclass · Accepted Answer

$8000 < N < 9000$

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(B) The pyramid has $18$ layers,with the top layer having $13$ balls on each side. Since it is a square-based pyramid,the number of balls in the $k$-th layer from the top is $k^2$ if we consider the top as the $13$-th layer of a full pyramid. Alternatively,the layers are $13^2, 14^2, 15^2, \dots, (13 + 18 - 1)^2 = 30^2$. The total number of balls $N$ is given by the sum of squares: $N = \sum_{k=13}^{30} k^2 = \sum_{k=1}^{30} k^2 - \sum_{k=1}^{12} k^2$. Using the formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$: $\sum_{k=1}^{30} k^2 = \frac{30(31)(61)}{6} = 5 	imes 31 	imes 61 = 9455$. $\sum_{k=1}^{12} k^2 = \frac{12(13)(25)}{6} = 2 	imes 13 	imes 25 = 650$. Therefore,$N = 9455 - 650 = 8805$. Since $8000 < 8805 < 9000$,the correct option is $B$.

Consider an incomplete pyramid of balls on a square base having $18$ layers,and having $13$ balls on each side of the top layer. Then,the total number $N$ of balls in that pyramid satisfies

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