Stair Patterns

Date: 02/27/2001 at 07:33:23
From: pierre aghajani
Subject: Need a formula, please

Could you help me with stair patterns made with matches?

The first step is made with four matches; the second step is made from 
10 matches; the third step is made from 18 matches; the fourth step is 
made from 28 matches.

How many matches would be needed to build 6, 10, and 50 steps? 

We must also find the number of steps in the largest stair pattern 
that can be made if we have 5000 matches.  

Thanks

Date: 02/27/2001 at 16:17:29
From: Doctor Jerry
Subject: Re: Need a formula, please

Hi  Pierre,

If you think about building these stairs each on the right side of the 
previous stair, then each of the bottom and top blocks in the new 
stack requires three matches, and the ones in between need only two.  
The number of in-between ones goes up by 1 each time

S1 = 4
S2 = S1+2*3     no in-between ones
S3 = S2+2*3+1*2
S4 = S3+2*3+2*2
S5 = S4+2*3+3*2
and so on.

So,

S1 = 4
S2 = 4+2*3
S3 = 4+4*3+1*2
S4 = 4+6*3+(1+2)*2
S5 = 4+8*3+(1+2+3)*2
and so on.

In S5, the 8 is 2(5-1)*3 and 1+2+3 is the first part of the sum of the 
integers, from 1 to 5-2. So,

Sn = 4+2(n-1)*3+2(n-1)(n-2)/2.

Using this formula you can figure out the number of matches  needed 
for 6, 10, and 50 steps.  

By experimenting a little you can show that S71 = 5254.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/