a In an arithmetic sequence there is a common difference between consecutive terms.
B
b In a geometric sequence there is a common ratio between consecutive terms.
C
c Make sure there is no common difference or common ratio between consecutive terms.
A
aFirst four terms: 4, 8, 12, 16
Equation: t(n)=4+4(n-1)
B
bFirst four terms: 4, 8, 16, 32
Equation: t(n)=4(2)^(n-1)
C
cExample sequence: 4, 8, 9, 14
Practice makes perfect
a In an arithmetic sequence there is a common difference separating consecutive terms. Since we have been given the first two terms, we can find this common difference and consequently the following two terms.
An arithmetic sequence in first term form is written in the following format.
t(n)=t(1)+m(n-1)
In this form, t(1) is the first term and m is the common difference. Examining the sequence, we can identify the first term as t(1)= 4 and we know that the common difference is m= 4. By substituting these values into the formula, we can write our equation.
t(n)= 4+ 4(n-1)
b In a geometric sequence there is a common ratio separating consecutive terms. Since we have been given the first two terms we can find this common ratio and consequently the following two terms.
A geometric sequence in first term form is written in the following format.
t(n)=t(1)a^(n-1)
In this form t(1) is the first term and a is the common ratio. Examining the sequence, we can identify the first term as t(1)= 4 and we know that the common ratio is m= 2. By substituting these values into the formula we can write our equation.
t(n)= 4( 2)^(n-1)
c As long as there is no common difference or a common ratio between consecutive terms, we can create any sequence we want. Below we see just one example.
