c Use the difference between t(1) and t(3) to find the rate of change.
D
d Since the sequence is geometric, we can calculate the common ratio by dividing t(3) by t(2).
E
e To go from t(7) to t(12), we have to add the common difference five times.
A
aExplicit: t(n)=3n-2
Recursive: t(n)=t(n-1)+3, t(0)=-2
B
bExplicit: t(n)=3(1/2)^(n-1)
Recursive: t(n)=1/2* t(n-1), t(0)=6
C
c t(n)=- 7n+24
D
d t(n)=6(1.2)^(n-1)
E
e t(4)=1620
Practice makes perfect
a The sequence can be either arithmetic or geometric. The difference between these sequences lies in the separation of consecutive terms.
Arithmetic: &Constant difference
&between consecutive terms
Geometric: &Constant factor
&between consecutive terms
Let's start by measuring the difference between consecutive terms.
Since the common difference is constant, this is an arithmetic sequence. Now we can determine the explicit and recursive rule for this sequence.
Explicit Rule
The explicit rule of an arithmetic sequence can be written in the following format.
&t(n)=d(n-1) + t(1)
&d=common difference
&t(1)=first term
We know the common difference and the sequence first term. By substituting these values into the equation, we can write the explicit rule.
To find a specific value with a recursive rule, we need to know that values previous value. When finding the explicit rule, we found that the zeroth term is -2. Therefore, we can write the rule as follows.
t(n)=t(n-1)+3, t(0)=-2
b Like in Part A, we will investigate what kind of a sequence it is. It looks geometric so we will try that first.
As we can see, this is a geometric sequence. Now we can determine the explicit and recursive rule for this sequence.
Explicit Rule
The explicit rule of a geometric sequence can be written in the following format.
&t(n)=t(1)r^(n-1) [0.3em]
&r=common factor
&t(1)=first term
We know that the common factor is r= 12 and the first term is t(1)= 3. With this information, we can write the equation.
t(n)= 3( 1/2)^(n-1)
Recursive Rule
Like we mentioned in Part A, to find a value using a recursive rule, we need to know its previous value. To write the recursive rule, we would like to know the zeroth value. Since the first term is 3 and the common difference is 12, the zeroth term must be 3Ă· 12=6. Therefore, we can write the rule as follows.
t(n)=1/2* t(n-1), t(0)=6
c Since the sequence is arithmetic, the rate of change is constant. Let's visualize this in the table.
To go from t(1) to t(3), we have to add the common difference twice. With this information, we can write the following equation.
17+2d=3
Let's solve the equation by performing inverse operations until d is isolated.
d Since the sequence is geometric, we know that there is a common ratio between consecutive terms. Let's visualize this in the table.
To go from t(2) to t(3), we have to multiply by the common ratio once. With this information, we can write the following equation.
7.2(r)=8.64 ⇔ r= 1.2
Before we can write the explicit rule we must also find the first term. To do that, we should divide t(2) by the common ratio.
Now we can write the explicit rule by substituting the common ratio r= 1.3 and the first term t(1)= 6 into the general equation of a geometric sequence.
t(n)= 6( 1.2)^(n-1)
e Let's visualize the arithmetic sequence in a table.
To go from t(7) to t(12), we have to add the common difference five times. With this information, we can write the following equation.
1056+5d=116
Let's solve the equation by performing inverse operations until d is isolated.
