Question

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

• A. $9000<N<10000$
• B. $8000<N<9000$
• C. $7000<N<8000$
• D. $10000<N<12000$

Answer 1

The correct option is B $8000<N<9000$

The top layer has $(13\times 13)$ balls the layer below it will have $(14\times 14)$ balls

We have $18$ layers

So the total number of balls

$N=(13\times 13)+(14\times 14)+.......(30\times 30)N={13}^2+{14}^2+.....{30}^2$

Sum of squares of first $n$ natural numbers is $\frac{n(n+1)(2n+1)}{6}$

$\therefore N=$ sum of first $30$ $-$ sum of first $12$

$N=\frac{30\times 31\times 61}{6}-\frac{12\times 13\times 25}{6}N=8805$

$\Rightarrow 8000<N<9000$

Hence, option B is correct.