a To determine if the sequence is arithmetic or geometric, we have to find out if there is a common difference or a common ratio between consecutive terms.
We have a common difference between consecutive terms which means this is an arithmetic sequence. The formula for an arithmetic sequence follows a certain format.
a_n= mn+a_0
In the formula, n is the term number, m is the common difference and a_0 is the zeroth term. We have already determined that the common difference is m= 3.
a_n= 3n+a_0
To find the zeroth term we can, for example, substitute a_1=1 in the formula and solve for a_0.
Now we can complete the equation.
a_n=3n+(- 2) ⇔ a_n=3n-2
b Again, to determine which kind of sequence this is, we will check if there is a common difference or common ratio between consecutive terms.
Since there is no common difference or common ratio between consecutive terms, this is neither an arithmetic or a geometric sequence.
c Like in Parts A and B, we will first determine if this is a geometric or an arithmetic sequence.
We have a common ratio between consecutive terms which means this is a geometric sequence. The formula for a geometric sequence follows a certain format.
a_n=a_0( b)^n
In the formula, n is the term number, b is the common ratio and a_0 is the zeroth term. We have already determined that the common ratio is b= 2.
a_n=a_0( 2)^n
To find the zeroth term, we can substitute a_1=2 in the formula and solve for a_0.
Now we can complete the equation.
a_n=1(2)^n ⇔ a_n=2^n
d Like in previous parts, let's determine if it's a geometric or an arithmetic sequence.
We have a common difference between consecutive terms which means this is an arithmetic sequence. Let's substitute the common difference m= 7 into the general formula for an arithmetic sequence.
a_n= 7n+a_0
To find the zeroth term we can substitute a_1=5 in the formula and solve for a_0.
