## anonymous one year ago 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

1. mathstudent55

D

2. freckles

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

3. anonymous

So what answer would it be using that?

4. mathstudent55

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,

5. freckles

so in general to find the (n+1)th term from the nth you do: $\frac{n(n+1)(2n+1)}{6}+(n+1)^2$

6. freckles

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

7. freckles

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

8. anonymous

so mathstudent is right?

9. freckles

yes (99+1)^2

10. freckles

I don't think he ever wrong :p

11. anonymous

he was right, thank you guys