The general form of a recursive formula is as follows.
A(n)=A(n-1)+d
In this formula, A(n) is the n^\text{th} term, A(n-1) is the previous term, and d is the common difference. We need to identify d, the common difference of the sequence.
3+2 ⟶5+2 ⟶7+2 ⟶9+2 ⟶...The common difference is +2. Using this information we can write the recursive formula for the sequence.
A(n)=A(n-1) +2; A(1)=3
Note, we need to include the value of the first term to make sure that our formula matches our sequence exactly.
Explicit Formula
The general form of an explicit formula is as follows.
A(n)=A(1)+(n-1)d
In this formula, A(n) is the n^\text{th} term, A(1) is the first term, n is the term number, and d is the common difference. Since A(1)=3 and d=2, the explicit formula can be written.
A(n)=3+(n-1)2
Let's simplify the equation.
The slope m of a linear function in slope-intercept form tells us the difference in y-values each time you move 1 step to the right in a coordinate system.
y= mx+b
Similarly, the recursive formula shows the common difference d between consecutive terms in a sequence.
A(n)=A(n-1)+ d
This means that we can think of an arithmetic sequence as y-coordinates in a linear function, and y-values in a linear function as an arithmetic sequence.
