Vocabulary
sequence?
An ordered list of numbers that has a pattern
Is this sequence arithmetic? If so, what is the common difference? If not, why is it not arithmetic? 1.5, 4, 6.5,...
Yes. The common difference is 2.5.
Is the following sequence geometric? If so, what is the multiplier? If not, explain why. t(n) = 54, 18, 6, ...
Yes. multiplier = 1/3
The explicit equation for an arithmetic sequence is t(n) = 3.5n +7. What is the recursive equation?
t(n+1) = t(n) + 3.5; t(1) = 10.5
What is the domain of any sequence?
n can be any natural number
arithmetic sequence?
A sequence where you add the same number each time
Is this sequence arithmetic? If so, what is the common difference? If not, why is it not arithmetic? 4, -5, -14, -25,...
No. This is not arithmetic, because you are not adding the same amount each time.
List the first three terms in the sequence, t(n) = 2(3)^n
6, 18, 54
The explicit equation for a geometric sequence is t(n) = 3.5(5)^n. What is the recursive equation?
t(n+1) = t(n)*5; t(1) = 17.5
What is the domain of the function f(x) = 1/x?
x = all real numbers except 0
geometric sequence?
A sequence where you multiply by the same number each time.
What is the explicit equation for the following arithmetic sequence? -50, -43, -36, -29, ...
t(n) = 7n - 57
What is the explicit equation for the following geometric sequence? 1.5, -6, 24 ...
t(n) = -0.375(-4)^n
What are the general explicit equations for arithmetic and geometric sequences?
Arithmetic: t(n) = (common difference)(n) + t(o); Geometric: t(n) = t(0)*multiplier^n
For the sequence t(n) = 3n + 7, can t(n) be 85? If so, what term number has this value? If not, explain why.
Yes, n = 26
common difference & multiplier
A common difference is an number that you add to a term in an arithmetic sequence to get the next term. A multiplier is a number that you multiply a term in a geometric sequence by to get the next term.
Mr. Stark averaged 123 total pins per game in his bowling league this season. He is taking bowling lessons and hopes to bring his average up by 8 pins each new season. Write an equation to represent the nth term of the sequence. Make sure to define your variables.
n = # of seasons; t(n) = bowling average; t(n) = 8n +115
Linda receives a joke in an E-mail that asks her to forward it on to four of her friends. She forwards it, then each of her friends forwards it to four of their friends, and so on. If the pattern continues, how many people will receive the E-mail on the ninth round of forwarding?
262,144 people
The recursive equation for an arithmetic sequence is t(n+1) = t(n) - 3 and t(2) = 5. What is the explicit equation?
t(n) = -3n + 11
For the sequence t(n) = 4(-3)^n, can t(n) be 324? If so, what term number has this value? If not, explain why.
Yes, n = 4
recursive equation
An equation that depends on knowing other terms in the sequence.
Write an explicit equation for an arithmetic sequence, t(n) = 17, 14, 11, ... and then determine t(50)
t(n) = -3n + 20; t(50) = -130
Write an explicit equation for a geometric sequence where t(2) = 16 and t(5) = 128.
t(n) = 4(2)^n
The recursive equation for a geometric sequence is t(n+1) = t(n)*0.5 and t(5) = 1/96. What is the explicit equation?
t(n) = (1/3)*(1/2)^n
For the sequence t(n) = 4000(1/2)^n, can t(n) be 200? If so, what term number has this value? If not, explain why.
No, t(n) cannot be 200. n will be a decimal between 4 and 5, which is not part of the domain of sequences.