How many cubes are in the 10th collection of cubes in this sequence?

1st- 1 cube

2nd- 4 cubes

3rd- 10 cubes.....

l

l

l

10th- ? cubes

Any help appreciated!

Printable View

- Mar 23rd 2009, 11:36 AMphillyfan09Elementary Math Problem
How many cubes are in the 10th collection of cubes in this sequence?

1st- 1 cube

2nd- 4 cubes

3rd- 10 cubes.....

l

l

l

10th- ? cubes

Any help appreciated! - Mar 23rd 2009, 12:38 PMGrandadTetrahedral numbers
Hello phillyfan09The sequence 1, 4, 10, ... is sometimes called Tetrahedral Numbers. Here's the answer to your question: Tetrahedral number - Wikipedia, the free encyclopedia.

Grandad - Mar 23rd 2009, 05:04 PMSoroban
Hello, phillyfan09!

Where did this problem come from?

It doesn't give enough information to allow us a good guess.

Lucky for us, Grandad recognized the sequence.

Quote:

How many cubes are in the 10th collection of cubes in this sequence?

. . $\displaystyle \begin{array}{cc}

\text{1st} & \text{1 cube} \\

\text{2nd} & \text{4 cubes} \\

\text{3rd} & \text{10 cubes} \\

\vdots & \vdots \\

\text{10th} & \text{? cubes} \end{array}$

Imagine a display of oranges, stacked in a triangular pyramid (tetrahedron).

. . Each horizontal level is an equilateral triangle.

$\displaystyle \begin{array}{ccccc}\text{Level} & \text{array} & \text{count} & \text{Total} \\ \hline \\[-4mm] 1 & o & 1 & 1\\ 2 & \begin{array}{c}o\\ [-3mm] oo\end{array} & 3 & 4\\ 3 & \begin{array}{c}o \\ [-3mm] oo \\[-3mm] ooo\end{array} & 6 & 10 \\ 4 & \begin{array}{c} o \\[-3mm] oo\\[-3mm]ooo\\[-3mm]oooo\end{array} & 10 & 20 \\\vdots & \vdots & \vdots & \vdots \end{array}$

And we want the 10th Total on this list.

You can crank out the numbers yourself, or you can use this formula:

. . $\displaystyle T(n) \;=\;\frac{n(n+1)(n+2)}{6}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

By the way, this formula answers the question:

How many gifts did My True Love Give To Me

. . during the Twelve Days of Christmas?

. . $\displaystyle \begin{array}{cccc}\text{Day} & \text{Gifts} & \text{Number} \\ \hline 1 & 1 & 1 \\ 2 &1+2 & 3 \\ 3 & 1+2+3 & 6 \\ 4 & 1+2+3+4 & 10 \\ \vdots & \vdots & \vdots \\ 12 & 1+2+\hdots+12 & 78 \\ \hline & \text{Total} & {\color{blue}364}\end{array}$

- Mar 24th 2009, 07:16 AMphillyfan09
Problem came from a college level elementary course...I believe professor made it up.

Thanks!