a Think about the name recursive formula. What does it imply?
B
b An arithmetic sequence in first term form is in the format t(n)=t(1)+m(n-1).
C
c Substitute n=50 into the formula from Part B.
D
d An arithmetic sequence in first term form is in the format t(n)=t(1)+m(n-1).
A
a Explicit equation
B
b t(n)=- 3+4(n-1)
C
c t(50)=193
D
d t(n)=3-1/3(n-1)
Practice makes perfect
a In an explicit equation, we can find a term by knowing its term number in the sequence. A recursive formula requires you to know the previous term to the term you want to calculate. If you want to calculate the 50^(th) term, with a recursive formula you would have to calculate the first 49 terms. Therefore, an explicit formula is more useful.
b An arithmetic sequence in first term form is 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 already see that the first term is - 3. By measuring the difference between consecutive terms, we can also determine the common difference.
By substituting t(1)= - 3 and m= 4, we can write the equation in first term form.
t(n)= - 3+ 4(n-1)
c To find the 50^(th) term we have to substitute n=50 in the equation and solve for n.
d Like in Part B, we have to find the first term and common difference or common ratio between consecutive terms.
From the diagram, we see that the first term is t(1)= 3 and the common difference is m= - 13. By substituting these values in the first term form of an arithmetic sequence, we can write our equation.
t(n)= 3 - 1/3(n-1)
