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