Unit 5 Flashcards
A blank is an ordered set of objects, like numbers, that follow a particular pattern.
sequence
Each element in a sequence is called a blank
term.
If there is a limited number of terms, this is a blank.
finite sequence
Sequences can also continue indefinitely, which is an blank.
infinite sequence
blank are used to denote that there is no last element in an infinite sentence.
Ellipses (…)
The blank of an infinite sequence is the set of positive integers.
domain
The term an
is called the
nth term, or the blank, of the sequence.
general term
Unlike sets, the terms in blank are in some type of “order”, such as increasing, decreasing, alphabetical or alternating.
sequences
An blank for a sequence allows you to find the value of any term in the sequence without determining all previous terms.
explicit formula
The nth term of the sequence can be found by raising 2
to the blank power.
nth
A blank is a function defined by multiple subsections.
a piecewise function
How to Find Explicit Formulas for a Sequence
Look for a pattern among the terms.
If the terms are fractions, look for a separate pattern among the numerators and denominators.
Look for a pattern among the signs of the terms.
Write a formula for an
in terms of n. Test the formula for n=1, n=2, and
n=3.
A blank defined sequence is one that is defined differently for different groups of n. For example, it may be defined one way for even values of n and another for odd values of n.
piecewise
An blank is a sequence {an} such that an <= an+1
for all integers n.
increasing sequence
An blank {an} is a sequence such that an < an+1
for all integers n.
strictly increasing sequence
A blank sequence {an} is a sequence such that an >= an+1
for all integers n.
decreasing
A blank is a sequence {an}such that an > an+1for all integers n.
strictly decreasing sequence
A blank is a sequence {an} that is either increasing or decreasing.
monotone sequence
A blank is a sequence that changes by a constant factor each year,
geometric sequence
The constant factor by which a geometric sequences increases or decreases is called the blank.
common ratio
A blank is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term.
geometric sequence
How to Identify a Geometric Sequence
Divide each term by the previous term.
Compare the quotients. If the quotients of any two consecutive terms are the same, a common ratio exists and the sequence is geometric.
How to Write Terms in a Geometric Sequence
Multiply the initial term by the common ratio to find the next term
Repeat the process until all four terms have been identified.
Write the terms separated by commas within brackets.
The nth term of a geometric sequence is given by what explicit formula:
an = a1r^(n-1)
An blank is a sequence that changes by a constant amount for each term.
arithmetic sequence
Each term increases or decreases by the same constant value, which is called the blank of the sequence.
common difference
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is a blank, called the common difference
constant,
The nth term of a arithmetic sequence is given by what explicit formula:
an = a1 + (n-1)d
How to Find a Term in an Arithmetic Sequence
Add the common difference to the first term to find the second term.
Add the common difference to the second term to find the third term.
Continue until all of the desired terms are identified.
Write the terms separated by commas within brackets.
How to Find Terms in an Arithmetic Sequence
Substitute the values given for a1, an, and d
into the formula an = a1 + (n-1)d
to solve for d.
Find a given term by substituting the appropriate values for a1, n, and d
into the formula an = a1 + (n-1)d.
How to find total terms in finite arithmetic sequence
Find the common difference d.
Substitute the common difference and the first term into an = a1 + (n-1)d.
Substitute the last term for an
and solve for n.
The sum of the terms of a sequence is called a blank.
series
The nth
partial sum of a sequence is the sum of a finite number of consecutive terms beginning with the first term. The notation blank
represents the partial sum.
Sn
Blank is used to represent series, and is often known as sigma notation because of the use of the Greek capital letter sigma
to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series.
Summation notation
The blank is set equal to the lower limit of summation, the number used to generate the first term in the series
index of summation
The number above sigma is called the blank, the number used to generate the last term in a series.
upper limit of summation
How to Calculate the Sum of a Finite Sequence
Identify the lower limit of summation.
Identify the upper limit of summation.
Substitute each value of k from the lower limit to the upper limit into the formula.
Add to find the sum.
An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first n
terms of an arithmetic sequence is blank
Sn = n(a1 + an)/2
A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first n
terms of a geometric sequence is represented as
Sn = (a1(1 - r^n))/1-r
How to find the sum of a geometric series
Identify a1, r, and n.
Substitute the values of into the formula Sn = (a1(1 - r^n))/1-r
Simplify to find Sn
A partial sum of an infinite series is a blank of the first k terms
finite sum
If the limit does not exist, we say that the infinite series blank
diverges
If a sequence converges, then it is blank.
bounded
