a What separates consecutive terms, a constant or a factor?
B
b Is this a line or a curve?
C
c Use the formula t(n)=a_1(r)^(n-1).
A
a Geometric sequence
B
b See solution.
C
c t(n)=2(5)^(n-1)
Practice makes perfect
a To determine what sequence this is we have to look at how it progresses. If a constant separates consecutive terms we have an arithmetic sequence, and if a factor separates consecutive terms we have a geometric sequence. Let's first determine if it's an arithmetic sequence.
The difference is not constant, and therefore this cannot be an arithmetic sequence. Let's test if it is a geometric sequence.
As we can see, there is a common ratio between consecutive terms. Therefore, this is a geometric sequence.
b From Part A we know that this is a geometric sequence with a common ratio of 5. This means the graph will be a curve rather than a straight line. For each step we go to the right, the value increases by 5 times the value before it.
c An explicit equation for a geometric sequence can be written as follows.
t(n)=a_1(r)^(n-1)
In this equation, a_1 is value of the first term when n=1, and r is the common ratio. Examining the sequence, we see that a_1= 2. Also, from Part A we identified the common ratio as r= 5. Now we can write the explicit equation.
t(n)= 2( 5)^(n-1)
