Question

The 99th pyramidal number is 328,350. That would make quite a stack of basketballs! What number must you add to 328,350 to get the 100th pyramidal number?

A. 1000
B. 55
C. 100
D. 10,000
E. 328,350

Answer 1

D

Answer 2

hmmm pyramid numbers...
well the first few pyramid numbers are: 1,5,14,30,55,91,140,....
And the n term can be given by :
\[\sum_{k=1}^{n} k^2=\frac{n(n+1)(2n+1)}{6}\]
so you have the 99th term which is given by:
\[\sum_{k=1}^{99} k^2=\frac{99(99+1)(2(99)+1)}{6}\]


anyways let me give you an example of what is going on here:
say we have the 3rd term which is given by:
\[\sum_{k=1}^{3} k^2=\frac{3(3+1)(2(3)+1)}{6}\]
which is 3(4)(7)/6=12(7)/6=2(7)=14
and we wanted to know what to add to 14 to get the next term
well notice we are adding squared numbers in that series above
so 14+4^2 to get the 4th term from the 3rd term

Answer 3

So what answer would it be using that?

Answer 4

1
1 + 4 = 5
5 + 9 = 14
14 + 16 = 30
30 + 25 = 55
Notice that to find the 2nd pyramidal number, you add 4 = 2^2 to the first one.
That means to find the 100th pyramidal number, you add 100^2 to the 99th number,

Answer 5

so in general to find the (n+1)th term from the nth you do:

\[\frac{n(n+1)(2n+1)}{6}+(n+1)^2\]

Answer 6

where n(n+1)(2n+1)/6 was the nth term

Answer 7

and (n+1)^2 was what you needed to add to get the next number (that I called (n+1)th

Answer 8

so mathstudent is right?

Answer 9

yes (99+1)^2

Answer 10

I don't think he ever wrong :p

Answer 11

he was right, thank you guys