a Which is the number you need to multiply a term by to get the next?
B
b Do the terms of the sequence approach a specific value?
C
c What happens to the values of the terms in the sequence as the n-values get higher?
D
d What input values are permitted?
A
aTerms: 4, 1, 0.25.
Equation: a_n=1024(0.25)^n.
B
b The terms would get smaller, but they would never become 0 or negative.
C
cDiagram:
The points approach the x-axes but do not intersect it.
D
dDomain Sequence: {1,2,3,4, ... }
Domain Function: x≥ 1
Practice makes perfect
a The sequence could either be arithmetic or geometric. Examining the difference between consecutive terms, we notice that the change is not linear which precludes an arithmetic sequence. Let's test if its geometric.
Since a common ratio separates consecutive terms, the sequence is in fact geometric. To find the next three terms, we can just continue the sequence.
The general form of a geometric sequence can be written in the following format.
a_n=a_0b^n
In this equation, a_0 is the zeroth term and b is the common ratio. By substituting the values of b and a_1, we can solve for the zeroth term.
b If we continue writing more terms in the series, we can use the equation a_n=1024* 0.25^n to see what will happen to the terms. As n gets larger, the term 0.25^n will get start to approach zero which means the sequence will get continuously smaller. However, the sequence will never be 0 as that would require us to multiply by 0
c From Part A, we have calculated the first six data point in the sequence. Let's plot these in a coordinate plane.
As we move to the right the sequence approaches the x-axis.
d The domain of the sequence is the set of all values n can take on. For this sequence, the domain is 1, 2, 3, 4..., which we also know as the natural numbers.
Domain Sequence: {1,2,3,4, ... }
The function with the same equation as the sequence can be written as y=1024(0.25)^x. Since the sequence begins with n=1 we let the function begin at x=1 as well. In this function, any real number greater than or equal to 1 is permitted. Therefore, the function will have the following domain.
Domain Function: x≥ 1
