Answer:
Yes all of the logs can be used in such a grouping.
We'll have <u>19 logs</u> in the bottom row (and have 19 rows)
========================================================
Explanation:
Let n be the number of layers. Let T(n) represent how many logs there are total.
- If you have 1 layer, then you have 1 log. So we can say that T(1) = 1.
- If there are n = 2 layers, then we have 1 log on top of two other logs, which leads to 1+2 = 3 logs total. This shows that T(2) = 3.
- If there are n = 3 layers, then we have 1 log on top of 2 which are on top of 3 at the very bottom. We have 1+2+3 = 6 logs total. So T(3) = 6.
- Having n = 4 layers gives 1+2+3+4 = 10 logs. So T(4) = 10
- Having n = 5 layers gives 1+2+3+4+5 = 15 logs. So T(5) = 15
This process continues for however long you like. The sequence 1,3,6,10,15 is the set of triangular numbers, since the logs form a triangular or pyramid shape when stacked this way. To get the nth value in the triangular number sequence, use this formula
T(n) = 0.5*n*(n+1)
---------------
For example, let's say we wanted to find how many logs there were in the fifth layer
T(n) = 0.5*n*(n+1)
T(5) = 0.5*5*(5+1)
T(5) = 15
This matches from earlier.
The purpose of bringing in a formula is to see if we can get T(n) = 190 for some integer n.
-------------------
T(n) = 0.5*n*(n+1)
190 = 0.5*n*(n+1)
190*2 = n(n+1)
380 = n^2+n
0 = n^2+n-380
n^2+n-180 = 0
Applying the quadratic formula will lead to the two solutions: n = -20 and n = 19. We ignore the negative value of n as we can't have a negative number of rows. Therefore, n = 19 is the only valid solution here.
If we wanted to stack 190 logs in this triangular pyramid fashion, then we'll need to have 19 rows. The bottom row will have 19 logs.
----------
Check:
1+2+3+4+5+6+7+8+9+10 = 55
11+12+13+14+15+16+17+18+19 = 135
55+135 = 190
Or you could say,
T(n) = 0.5*n*(n+1)
T(19) = 0.5*19*(19+1)
T(19) = 190
Either way, the answer is confirmed.
