2.3 Exercises
For # 1 – 8, determine if each of the following sequences is arithmetic or geometric. Then write recursive and explicit formulas/rules for each using subscript notation.
1) [latex]–10, –20, –30, –40,[/latex] …
2) [latex]5, –15, 45, –135,[/latex] …
3) [latex]2, 3.0, 5.8, 8.6,[/latex] …
4) [latex]\frac{1}{12},\frac{1}{6},\frac{1}{3},\frac{1}{2}[/latex],… (hint: find a common denominator first)
5) [latex]\frac{1}{4},\frac{3}{4},1\frac{1}{4}[/latex],…
6) [latex]\frac{1}{3},\frac{1}{6},\frac{1}{12}[/latex],…
7) [latex]3\frac{11}{12},3\frac{7}{12},3\frac{1}{4},2\frac{11}{12},2\frac{7}{12}[/latex]…
(Observe [latex]3\frac{1}{4}=3\frac{4}{12}[/latex])
8) [latex]1\frac{2}{3},\frac{4}{9},\frac{8}{27}[/latex],…
9) Miguel’s annual income has been increasing by the same amount every year. In the first year his income was $50,000. In the 6th year it will be $58,000. In what year will his income be $66,000.
10) One hundred kilograms of a toxic chemical was dumped illegally into a clean reservoir. A filter can remove 20% of the chemical still present each week (so that 80% of the previous amount remains). How much of the chemical will remain in the water after 20 weeks?
An ordered list of objects (e.g., numbers) usually with a certain pattern.
A pattern or sequence in which the same number is added to each term to get the next term.
A sequence in which the same number is multiplied by each term to get the next term. The number that we are multiplying by is called the common ratio, denoted r.
A recursive formula for a sequence identifies initial term(s)/number(s) in the sequence and then gives a rule for finding each subsequent term in the sequence using one or more previous terms.
A formula for the nth term of a sequence in which we may plug in the value of n to determine the nth term without using previous terms.
A notation for terms of a sequence that uses subscripts to indicate the position of the term in the sequence (e.g., a_n read "a sub n" denotes the nth term of the sequence).
