a Let's calculate the first four terms for Wade using the given information.
n
4(3)^n
t(n)
1
4(3)^1
12
2
4(3)^2
36
3
4(3)^3
108
4
4(3)^4
324
Let's also calculate the first four terms for Dwayne.
n
12(3)^(n-1)
t(n)
1
12(3)^(1-1)
12
2
12(3)^(2-1)
36
3
12(3)^(3-1)
108
4
12(3)^(4-1)
324
As we can see, both equations give the same sequence. Therefore, both forms of the equation are correct.
b A geometric sequence can be written in two ways. Either you start with the first term, or you start with the zero term.
First term:& t(n)= a_1b^(n-1)
& a_1= first term [0.8em]
Zeroth term:& t(n)= a_0b^n
& a_0= zeroth term
As we can see, Dwayne has written the equation in first-term form.
c Let's calculate the first four terms for Wade.
n
9.1+1.2n
t(n)
1
9.1+1.2( 1)
10.3
2
9.1+1.2( 2)
11.5
3
9.1+1.2( 3)
12.7
4
9.1+1.2( 4)
13.9
Let's also calculate the first four terms for Dwayne.
n
10.3+1.2(n-1)
t(n)
1
10.3+1.2( 1-1)
10.3
2
10.3+1.2( 2-1)
11.5
3
10.3+1.2( 3-1)
12.7
4
10.3+1.2( 4-1)
13.9
As we can see, both equations give the same sequence. Therefore, both forms of the equation are correct.
d Dwayne calls the equation first term form, because his equation starts with the first term of the sequence. We have to subtract 1 from n because if we substitute n=1 into the first term form we are left with a_1.
