Question 986170: Jack is building a pyramid by leaning cards against each other. He starts with 2 cards. This is a "1-level" pyramid and it uses 2 cards.
By making the base wider, and putting a 'floor' between each level, he can make higher pyramids... A '2 level' pyramid uses 7 cards, a '3 level' pyramid uses 15 cards and so on.
Write an equation showing how many cards are needed (c) for an 'n-level' pyramid
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Draw out a few cases. I drew out pyramids with 1 level, 2 levels, 3 levels, 4 levels, and 5 levels. Basically I am showing below cases n = 1 through n = 5
The list goes on, but eventually we run out of room, so it's a good place to stop here. We have enough to see a pattern forming.
Notice the sequence of cards needed: 2, 7, 15, 26, 40, ...
The jump from 2 to 7 is +5
The jump from 7 to 15 is +8
The jump from 15 to 26 is +11
The jump from 26 to 40 is +14
We have a list of common differences: 5, 8, 11, 14
Each common difference is separated by 3.
5+3 = 8
8+3 = 11
11+3 = 14
and so on...
Since we have constant 2nd differences (all 3), this means a quadratic polynomial models the original sequence.
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Pair up this list {1,2,3,4,5} with {2, 7, 15, 26, 40} to get this list of ordered pairs
(1,2), (2,7), (3,15), (4,26), (5,40)
we only need 3 ordered pairs. Let's just work with the first three ordered pairs. We will work with the general quadratic form y = ax^2+bx+c
Plug in (x,y) = (1,2) into y = ax^2+bx+c to get
y = ax^2+bx+c
2 = a*1^2+b*1+c
a+b+c = 2
Plug in (x,y) = (2,7) into y = ax^2+bx+c to get
y = ax^2+bx+c
7 = a*2^2+b*2+c
4a+2b+c = 7
Plug in (x,y) = (3,15) into y = ax^2+bx+c to get
y = ax^2+bx+c
15 = a*3^2+b*3+c
9a+3b+c = 15
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After plugging in those (x,y) pairs and simplifying, we have this system of equations
a+b+c = 2
4a+2b+c = 7
9a+3b+c = 15
There are a number of ways to solve. I'm going to use a calculator. Specifically the TI83's inverse matrix feature.
Doing so essentially solves for (a,b,c) simultaneously. The calculator states that
a = 1.5
b = 0.5
c = 0
So the nth term equation is c = 1.5n^2 + 0.5n where n is the level number and c is the number of cards needed to make the pyramid that tall.
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For instance, if n = 3, then we should get c = 15. Let's see if that's true.
c = 1.5n^2 + 0.5n
c = 1.5*3^2 + 0.5*3
c = 1.5*9 + 0.5*3
c = 13.5 + 1.5
c = 15
we definitely get c = 15. So we need 15 cards for a 3 level pyramid. This confirms the equation is true for n = 3. Try other values of n (like n = 1 or n = 2) to generate other values of c. This is a good way to confirm we have the correct nth term formula.
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