Algebra 2
A sequence is of pyramids is created where the first pyramid has a height of 6 inches and a base area of 10 inches, for a volume of 20 cubic inches. The height of the next pyramid is twice the height of the previous pyramid. The input of the sequence is the position of the pyramid and the output is the volume of the pyramid. Is the sequence explicit or recursive? Is the sequence arithmetic, geometric, or neither?
I don't understand and don't know where to start from to find the answer out please help
Confusing wording:
"he first pyramid has a height of 6 inches and a base area of 10 inches"
since you tell us the volume is 20 inches^2 , the base are must be 10 inches^2
I also will assume that the base does not change
volume of 1st pyramid = (1/3)(10)(6) = 20
volume of 2nd pyramid = (13)(10)(12) = 40
volume of 3rd ---- = (1/3)(10)(24) = 80
20, 40, 80, ... clearly show a geometric sequence
I could write the volume either as an explicit expression or a recursive
explicit: ----> V(n) = 10(2)^n
recursive ---> V(n) = 2V(n-1), n > 1, V(1) = 20
but where do i get the 1/3 from?
To determine whether the sequence is explicit or recursive, we need to understand the difference between the two:
- An explicit sequence is one that can be defined directly in terms of the position of an element in the sequence. In other words, the formula for finding any term in the sequence can be written without reference to the previous terms.
- A recursive sequence is one that defines each term in the sequence in terms of the previous terms. In other words, the formula for finding any term in the sequence depends on the values of one or more preceding terms.
In this case, we can determine the formula for the volume of each pyramid by analyzing the given information.
We are told that the height of the first pyramid is 6 inches, and its volume is 20 cubic inches. From this, we can deduce the base area of the first pyramid using the formula for the volume of a pyramid:
Volume = (1/3) * Base Area * Height
Substituting the given values, we have:
20 = (1/3) * Base Area * 6
Simplifying this equation, we find:
Base Area = 10 inches
We are also told that each subsequent pyramid's height is twice the height of the previous pyramid. This means that the sequence of heights is: 6, 12, 24, ...
Given that the base area remains constant at 10 square inches for all pyramids, we can use the formula for the volume of a pyramid to calculate the volume of any pyramid in the sequence:
Volume = (1/3) * Base Area * Height
Since the height of each pyramid is determined by doubling the previous height, we can express the height of the nth pyramid as follows:
Height_n = 6 * (2^(n-1))
where n represents the position of the pyramid in the sequence.
By substituting the base area and height formulas into the volume formula, we can explicitly express the volume of the nth pyramid in terms of its position:
Volume_n = (1/3) * 10 * 6 * (2^(n-1))
Therefore, the sequence is explicit because we can directly calculate the volume of any pyramid using the formula derived above.
Now let's determine if the sequence is arithmetic, geometric, or neither:
An arithmetic sequence is one in which the difference between consecutive terms is constant. For example, the sequence 1, 4, 7, 10, ... has a common difference of 3.
A geometric sequence is one in which each term is obtained by multiplying the previous term by a constant factor. For example, the sequence 2, 6, 18, 54, ... has a common ratio of 3.
In this case, we can see that the sequence of volumes is obtained by multiplying the previous volume by a constant factor of 2 (from the expression 2^(n-1)). Therefore, the sequence is geometric.