1. Sequences and series

Question 1

number sequence

Answer 1

an ordered list of numbers defined by a rule

Question 2

members or terms

Answer 2

the numbers in a sequence

Question 3

an infinite sequence

Answer 3

a sequence which continues forever

Question 4

a finite sequence

Answer 4

a sequence which terminates

Question 5

sequences can be defined by:

Answer 5

• listing terms
• using words
• using an explicit formula (Uˇn=…)
• a graphical representation (each number is a dot)
• using a recursive formula (have to go steps back) (Uˇ1=… Uˇn+1=…)

Question 6

arithmetic sequence

Answer 6

a sequence in which each term differs from the previous one by the same fixed number (common difference)

Question 7

Arithmetic sequence is also referred to as…

Answer 7

..arithmetic progression.

Question 8

If d<0 the sequence is…

Answer 8

…decreasing and the other way around.

Question 9

Why the name arithmetic?

Answer 9

Because a number in the sequence is an arithmetic mean of the numbers on either side of it.

Question 10

What is this formula called:
Uˇn=Uˇ1+(n-1)d

Answer 10

It’s called the general term (n) formula.

Question 11

geometric sequence

Answer 11

a sequence is geometric if each term can be obtained from the previous one by multiplying by the same non-zero constant - this is called geometric progression

Question 12

Why the name geometric?

Answer 12

b^2=ac (geometric mean)

Question 13

What is the general term formula for geometric sequences<’

Answer 13

Uˇn=Uˇ1*r^(n-1)

Question 14

What is “r” in Uˇ(n+1)/Uˇn=r called?

Answer 14

the common ratio

Question 15

a series

Answer 15

the sum of terms of a sequence

Question 16

corresponding series for the finite sequence {Uˇn}

Answer 16

Uˇ1+Uˇ2+…+Uˇn

Question 17

The series (sum) of a finite sequence is…

Answer 17

…always a finite, real number.

Question 18

corresponding series for the infinite sequence {Uˇn}

Answer 18

Uˇ1+Uˇ2+…Uˇn+…

Question 19

The series (sum) of an infinite sequence is…

Answer 19

…in many cases impossible to calculate, but in other cases does converge to a finite number (rarely).

Question 20

How is the sigma notation with determined k, max value and form read?

Answer 20

The sum of all numbers of the form Uˇk where k is 1, 2, 3, etc. up to n.

Question 21

properties of sigma notation:

Answer 21

1) sigma (aˇk+bˇk) = sigma (aˇk) + sigma (bˇk)
2) sigma (caˇk) where c is constant = csigma (aˇk)
3) sigma (c) = n*c

Question 22

arithmetic series

Answer 22

the sum of the terms of an arithmetic sequence

Question 23

sum of a finite arithmetic sequence (formula)

Answer 23

Sˇn=[n(Uˇ1+Uˇn)]/2

Question 24

geometric sequence

Answer 24

the sum of the terms of a geometric sequence

Question 25

sum of a finite geometric sequence (formula)

Answer 25

Sˇn=[Uˇ1(r^n-1)]/(r-1), when r is not 1(otherwise Sˇn=n*Uˇ1)

Question 26

If |r|>1 (r>-1, r>1) then the series is set to be…

Answer 26

…divergent.

Question 27

In divergent series the sum is…

Answer 27

…infinitely large.

Question 28

If |r|<1 (-1<r<1) then the series is set to be…

Answer 28

…convergent.

Question 29

In convergent series, as n becomes very large, r^n…

Answer 29

…approaches 0.

Question 30

An infinite, (convergent) geometric sequence of the form sigma notation (k=1, n=infinity, Uˇk=Uˇ1^k-1) will converge to the sum…

Answer 30

…S=(Uˇ1)/(1-r)