Pyramid

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ARITHMETIC SEQUENCES<BR>

THE HUMAN PYRAMID

pyramid

Corporations and bureaucracies have chains of command that are structured like a pyramid with most of us on the bottom and the more powerful few at the top. Above is an illustration of one such possible corporate pyramid. The number of people in the rows of the pyramid form an arithmetic sequence:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

How many people are there altogether?

Before trying to answer this question, look at the two figures below.
The figure below suggests a shortcut for finding the number of people in a pyramid with 6 rows.

pyramid6


  1   2   3   4   5   6

 +6  +5  +4  +3  +2  +1

 __  __  __  __  __  __

  7   7   7   7   7   7
 
= 42


42/2 = 21 people

Find the number of people in the pyramid of 16 rows
by using the shortcut suggested by this pattern:


  1   2   3   4   5   6   7 ... 16 

+16 +15 +14 +13 +12 +11 +10 ... +1

___ ___ ___ ___ ___ ___ ___     ___

 17  17  17  17  17  17  17     17

Check your answer by adding the numbers in order from smallest to largest on a calculator:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16.
Find the sum of the 12 terms shown of this arithmetic sequence by adding them in order from left to right.

1 3 5 7 9 11 13 15 17 19 21 23
Find the same sum by using the shortcut suggested by this pattern:
```
      1   3   5 ...

    +23 +21 +19 ...

    ___________

     24  24  24 ...
```
Build your own human pyramid with the mathlet above.
To clone more people, click where you want one and press the "m" or the "w" key.
Drag them around for precise placement.
Change their colors in the text boxes. Then press Enter.
To delete a figure, click on it. Then press The Escape or Backspace key.
Create a human pyramid with 10 rows.
Place men and women in alternate rows
so that each row contains only men or only women.
How many people will you need?
How many men will you need?
How many women will you need?
If you know how many rows you need in your corporate pyramid,
how can you quickly find how many people you need?
Describe how in English. Then write a formula.

Can You Transfer Your Ideas To Other Situations?

As every bowler knows, the pins are set up in four rows with one pin in the first row,
two pins in the second row, three pins in the third, and four in the last.
Suppose that a gigantic set of pins was set up in 20 rows,
the first row having 1 pin and each successive row having 1 more,
so that the last row had 20 pins.
How many pins would you have to knock over to make a strike?
If pins were set up in the same fashion in 100 rows,
how many pins would you have to knock over to make a strike?

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Last Modified 12/21/08 10:08 AM

Pyramid

THE HUMAN PYRAMID

Can You Transfer Your Ideas To Other Situations?