A produce store has stacked apples in the shape of a triangular pyrimid. The first layer on top contains a single apple. The second layer contains 3 apples arranged in a triangle. The third layer contains 1+2+3=6 apples. Using algebra, convert your formula, k(k+1)/2, into the form ak^2+bk+c. That is, find the three values of a, b, and c.
The number of apples in each layer are called "triangular" numbers, that is,
1, 3, 6, 10, 15, ....
think of the balls in a game of billiards and the total if you rack them up.
So the number of apples in each layer working down from the top are
1, 3, 6, 10, 15, ....
they gave you the formula producing these numbers, e.g for level 4, k=4
and the number is 4(5)/2 = 10, the fourth number in my list
k(k+1)/2 = (k^2 + k)/2 or (1/2)k^2 + (1/2)k + 0
so for the given question, a = 1/2, b = 1/2, and c = 0
I anticipate the next question to be to find the number of apples for a given
number of layers.
168
The triangular numbers are the numbers in each layer. Can you extend their formula to give you the number apples in a pyramid of n layers? That is, the formula for pyramidal numbers? That would be
1: 1
2: 1+3
3: 1+3+6
4: 1+3+6+10
...
165 apples, not sure
130
To convert the formula k(k+1)/2 into the form ak^2 + bk + c, we need to expand and simplify it.
The formula k(k+1)/2 represents the sum of the first k positive integers. Let's expand and simplify it step by step:
Step 1: Expand k(k+1) by multiplying:
k(k+1) = k^2 + k
Step 2: Divide the result by 2:
k(k+1)/2 = (k^2 + k)/2
Step 3: Rewrite the expression in the desired form, ak^2 + bk + c:
(k^2 + k)/2 = ak^2 + bk + c
To determine the coefficients a, b, and c, we need to match the expanded expression with the form ak^2 + bk + c.
In this case, we see that a = 1/2 since the coefficient of k^2 is 1/2.
To determine b and c, we can rewrite the expression (k^2 + k)/2 as follows:
k^2/2 + k/2 = (1/2)k^2 + (1/2)k + 0
Comparing the coefficients, we conclude that b = 1/2 and c = 0.
Therefore, the formula k(k+1)/2 can be written as (1/2)k^2 + (1/2)k + 0 in the form ak^2+bk+c.