A sequence is an ordered set of numbers with a well-defined rule for getting from one number to the next.

Define " term " in relation to sequences.

A term is one of the numbers in a sequence.

A series is the sum of the terms in a sequence .

Define the term n th term formula .

The n th term formula is a formula that allows you to find any term in a sequence by substituting the term number (i.e. the value of n ).

True or False? A GDC can use sigma notation .

True. A GDC can use sigma notation . Be sure to know how this feature works on your GDC.

What is an arithmetic sequence ?

An arithmetic sequence is a sequence where the difference between consecutive terms is constant .

True or False? An arithmetic sequence can only be increasing .

False. An arithmetic sequence can be increasing (positive common difference) or decreasing (negative common difference).

What is an arithmetic series ?

An arithmetic series is the sum of the terms in an arithmetic sequence .

True or False? Simultaneous equations are often needed to solve arithmetic sequence problems.

True. Simultaneous equations are often needed to solve arithmetic sequence problems.

Define consecutive terms .

Consecutive terms are terms that follow each other directly in a sequence.

True or False? The sum of an arithmetic series always increases as more terms are added.

False. The sum of an arithmetic series can decrease if the common difference is negative .

What is a geometric sequence ?

A geometric sequence is a sequence where there is a common ratio between consecutive terms.

Define the term common ratio .

The common ratio is the constant factor by which each term in a geometric sequence is multiplied to get the next term.

What is a geometric series ?

A geometric series is the sum of the terms in a geometric sequence .

True or False? Logarithms are sometimes needed to solve geometric sequence problems.

True. Logarithms are sometimes needed to solve geometric sequence problems.

What is an alternating sequence ?

An alternating sequence is a sequence where the terms alternate between positive and negative values . This happens for a geometric sequence when the common ratio r is negative .

What type of sequence is appropriate when a quantity changes by a fixed amount repeatedly?

An arithmetic sequence is appropriate when a quantity changes by a fixed amount repeatedly.

True or False? Simple interest is when an initial investment is made and then a percentage of the initial investment (and only of the initial investment) is added to this amount on a regular basis.

True. Simple interest is when an initial investment is made and then a percentage of the initial investment (and only of the initial investment) is added to this amount on a regular basis.

What type of sequence is appropriate when a quantity changes by a fixed percentage repeatedly?

A geometric sequence is appropriate when a quantity changes by a fixed percentage repeatedly.

Define compound interest .

Compound interest is when an initial investment is made and then interest is paid on the initial amount and on the interest already earned on a regular basis.

True or False? Calculating simple interest can be modelled using geometric sequences .

False. Calculating simple interest can be modelled using arithmetic sequences .

True or False? Geometric sequences can be used to solve compound interest problems.

True. Geometric sequences can be used to solve compound interest problems. (The financial applications feature of your GDC should also be able to solve compound interest problems.)

True or False? Exam questions will always explicitly state whether to use sequences and series methods.

False. Exam questions won't always explicitly state whether to use sequences and series methods. You may need to deduce from the context of the question that such methods are necessary.

What should you look for to identify a " hidden " geometric sequence question ?

To identify a " hidden " geometric sequence question , look for situations where a given amount is changing by a percentage or multiple .

True or False? The spread of a virus can be modelled using sequences and series .

True. The spread of a virus can be modelled using sequences and series , often using geometric sequences .

What is simple interest ?

Simple interest is interest paid only on the initial investment , not on accumulated interest.

True or False? Compound interest is interest that is paid both on the initial investment and on any interest that has already been paid .

True. Compound interest is interest that is paid both on the initial investment and on any interest that has already been paid .

True or False? Compound interest results in higher returns than simple interest over time.

True. Compound interest results in higher returns than simple interest over time.

What is depreciation ?

Depreciation is when the value of something decreases over time.

What is the nominal annual rate of interest ?

The nominal annual rate of interest is the stated interest rate without taking into account compounding periods .

True or False? Compounding monthly means k = 12 in the compound interest formula.

True. Compounding monthly means k = 12 in the compound interest formula, because there are 12 months in a year.

IBMathsDPAnalysis & Approaches (AA)SLFlashcards1. Number & AlgebraSequences & Series

Sequences & Series (DP IB Analysis & Approaches (AA))

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What is a sequence?
A sequence is an ordered set of numbers with a well-defined rule for getting from one number to the next.
Define "term" in relation to sequences.
A term is one of the numbers in a sequence.
True or False?
The terms of a sequence are always referred to by letters with a subscript.
False.
While terms are often referred to by letters with a subscript (e.g. $u_{1}, u_{2}, u_{3}, . . .$ ), this is not always the case.
What is a series?
A series is the sum of the terms in a sequence.
What does $S_{n}$ represent?
$S_{n}$ represents the sum of the first n terms in a series.
Define the term nth term formula.
The nth term formula is a formula that allows you to find any term in a sequence by substituting the term number (i.e. the value of n).
What does the symbol $\sum$ stand for in sigma notation?
The symbol $\sum$ stands for "sum" in sigma notation.
What is the purpose of the expression to the right of $\sum$ in sigma notation?
The expression to the right of $\sum$ in sigma notation tells you what is being summed.
E.g. $\sum_{r = 1}^{4} 2 r - 1$ means to sum up the terms in the sequence generated by the expression $2 r - 1$ for the specified values of $r$ .
What do the limits above and below $\sum$ indicate in sigma notation?
The limits above and below $\sum$ in sigma notation tell you which terms you are summing.
E.g. For $\begin{array}{rcl} \sum_{r = 1}^{4} 2 r - 1 \end{array}$ , you are summing the terms between 1 and 4. $\begin{array}{rcl} \sum_{r = 1}^{4} 2 r - 1 & = & (2 (1) - 1) + (2 (2) - 1) + (2 (3) - 1) + (2 (4) - 1) \\ = & 1 + 3 + 5 + 7 \\ = & 16 \end{array}$
True or False?
The limits in sigma notation always start with 1.
False.
The limits in sigma notation don't always start with 1.
E.g. $\sum_{k = 3}^{6} k^{2} = 3^{2} + 4^{2} + 5^{2} + 6^{2}$
True or False?
A GDC can use sigma notation.
True.
A GDC can use sigma notation.
Be sure to know how this feature works on your GDC.
What is an arithmetic sequence?
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
Define the term common difference.
The common difference is the constant difference between consecutive terms in an arithmetic sequence.
The common difference is usually denoted by $d$ .
What is the nth term formula for an arithmetic sequence?
The nth term formula for an arithmetic sequence is $u_{n} = u_{1} + (n - 1) d$
Where:
- $u_{n}$ is the nth term
- $u_{1}$ is the first term
- $n$ is the term number
- $d$ is the common difference
This formula is in the exam formula booklet.
True or False?
An arithmetic sequence can only be increasing.
False.
An arithmetic sequence can be increasing (positive common difference) or decreasing (negative common difference).
What is an arithmetic series?
An arithmetic series is the sum of the terms in an arithmetic sequence.
What is the formula for the sum of the first n terms of an arithmetic series in terms of $u_{1}$ , $n$ and $d$ ?
The formula for the sum of the first n terms of an arithmetic series in terms of $u_{1}$ , $n$ and $d$ is $S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d)$
Where:
- $S_{n}$ is the sum of the first n terms
- $u_{1}$ is the first term
- $n$ is the term number
- $d$ is the common difference
This formula is in the exam formula booklet.
What is the formula for the sum of the first n terms of an arithmetic series in terms of $u_{1}$ and $u_{n}$ ?
The formula for the sum of the first n terms of an arithmetic series in terms of $u_{1}$ and $u_{n}$ is $S_{n} = \frac{n}{2} (u_{1} + u_{n})$
Where:
- $S_{n}$ is the sum of the first n terms
- $u_{1}$ is the first term
- $u_{n}$ is the nth term
This formula is in the exam formula booklet.
True or False?
Simultaneous equations are often needed to solve arithmetic sequence problems.
True.
Simultaneous equations are often needed to solve arithmetic sequence problems.
Define consecutive terms.
Consecutive terms are terms that follow each other directly in a sequence.
What is the relationship between $u_{n}$ and $u_{n + 1}$ in an arithmetic sequence?
In an arithmetic sequence, $u_{n + 1} = u_{n} + d$ , where $d$ is the common difference.
$u_{n}$ and $u_{n + 1}$ are consecutive terms.
True or False?
The sum of an arithmetic series always increases as more terms are added.
False.
The sum of an arithmetic series can decrease if the common difference is negative.
What is a geometric sequence?
A geometric sequence is a sequence where there is a common ratio between consecutive terms.
Define the term common ratio.
The common ratio is the constant factor by which each term in a geometric sequence is multiplied to get the next term.
What is the nth term formula for a geometric sequence?
The nth term formula for a geometric sequence is $u_{n} = u_{1} r^{n - 1}$
Where:
- $u_{1}$ is the first term
- $n$ is the term number
- $r$ is the common ratio
This formula is in the exam formula booklet.
True or False?
A geometric sequence can only be increasing.
False.
A geometric sequence can be increasing ( $r > 1$ ), decreasing ( $0 < r < 1$ ), or alternating between positive and negative ( $r < 0$ ).
What is a geometric series?
A geometric series is the sum of the terms in a geometric sequence.
State the formula that should be used for the sum of the first n terms of a geometric series when r > 1.
When r > 1, the formula that should be used for the sum of the first n terms of a geometric series is $S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1}$
Where:
- $u_{1}$ is the first term
- $n$ is the term number
- $r$ is the common ratio
This formula is in the exam formula booklet.
State the formula that should be used for the sum of the first n terms of a geometric series when r < 1.
When r < 1, the formula that should be used for the sum of the first n terms of a geometric series is $S_{n} = \frac{u_{1} (1 - r^{n})}{1 - r}$
Where:
- $u_{1}$ is the first term
- $n$ is the term number
- $r$ is the common ratio
This formula is in the exam formula booklet.
True or False?
Logarithms are sometimes needed to solve geometric sequence problems.
True.
Logarithms are sometimes needed to solve geometric sequence problems.
What is an alternating sequence?
An alternating sequence is a sequence where the terms alternate between positive and negative values.
This happens for a geometric sequence when the common ratio r is negative.
What is the relationship between $u_{n}$ and $u_{n + 1}$ in a geometric sequence?
In a geometric sequence, $u_{n + 1} = u_{n} \times r$ , where r is the common ratio.
$u_{n}$ and $u_{n + 1}$ are consecutive terms.
True or False?
The sum to infinity can be calculated for any geometric series.
False.
The sum to infinity can not be calculated for any geometric series.
The sum to infinity can only be calculated if the geometric series converges. This happens if $|r| < 1$ , where $r$ is the common ratio of the geometric series.
What is the formula for calculating the sum to infinity of a geometric series?
The formula for calculating the sum to infinity of a geometric series is $S_{\infty} = \frac{u_{1}}{1 - r}, |r| < 1$
Where:
- $u_{1}$ is the first term of the geometric series
- $r$ is the common ratio of the geometric series
The formula is only valid if the convergence condition $|r| < 1$ is satisfied.
This formula is in the exam formula booklet.
True or False?
$|r| < 1$ is the same thing as $- 1 < r < 1$ .
True.
$|r| < 1$ is the same thing as $- 1 < r < 1$ .
Those are two equivalent ways of writing the same inequality.
What type of sequence is appropriate when a quantity changes by a fixed amount repeatedly?
An arithmetic sequence is appropriate when a quantity changes by a fixed amount repeatedly.
True or False?
Simple interest is when an initial investment is made and then a percentage of the initial investment (and only of the initial investment) is added to this amount on a regular basis.
True.
Simple interest is when an initial investment is made and then a percentage of the initial investment (and only of the initial investment) is added to this amount on a regular basis.
What type of sequence is appropriate when a quantity changes by a fixed percentage repeatedly?
A geometric sequence is appropriate when a quantity changes by a fixed percentage repeatedly.
Define compound interest.
Compound interest is when an initial investment is made and then interest is paid on the initial amount and on the interest already earned on a regular basis.
True or False?
Calculating simple interest can be modelled using geometric sequences.
False.
Calculating simple interest can be modelled using arithmetic sequences.
True or False?
Geometric sequences can be used to solve compound interest problems.
True.
Geometric sequences can be used to solve compound interest problems.
(The financial applications feature of your GDC should also be able to solve compound interest problems.)
True or False?
Exam questions will always explicitly state whether to use sequences and series methods.
False.
Exam questions won't always explicitly state whether to use sequences and series methods.
You may need to deduce from the context of the question that such methods are necessary.
What should you look for to identify a "hidden" geometric sequence question?
To identify a "hidden" geometric sequence question, look for situations where a given amount is changing by a percentage or multiple.
True or False?
The spread of a virus can be modelled using sequences and series.
True.
The spread of a virus can be modelled using sequences and series, often using geometric sequences.
What is simple interest?
Simple interest is interest paid only on the initial investment, not on accumulated interest.
True or False?
Compound interest is interest that is paid both on the initial investment and on any interest that has already been paid.
True.
Compound interest is interest that is paid both on the initial investment and on any interest that has already been paid.
True or False?
Compound interest results in higher returns than simple interest over time.
True.
Compound interest results in higher returns than simple interest over time.
State the formula for compound interest.
The formula for compound interest is $F V = P V \times {(1 + \frac{r}{100 k})}^{k n}$
Where:
- $F V$ is future value
- $P V$ is present value
- $n$ is the number of years
- $k$ is the number of compounding periods per year
- $r %$ is the nominal annual rate of interest
This equation is in the exam formula booklet. The financial applications on your GDC can also calculate compound interest without having to use this formula.
What is depreciation?
Depreciation is when the value of something decreases over time.
True or False?
The formula for compound depreciation is the same as for compound interest.
False.
The formula for compound depreciation is similar to the one for compound interest.
For compound depreciation the formula is $F V = P V \times {(1 - \frac{r}{100})}^{n}$
Where:
- $F V$ is future value
- $P V$ is present value
- $n$ is the number of years
- $r %$ is the annual rate of depreciation
This equation is not in the exam formula booklet.
What is the nominal annual rate of interest?
The nominal annual rate of interest is the stated interest rate without taking into account compounding periods.
True or False?
Compounding monthly means k = 12 in the compound interest formula.
True.
Compounding monthly means k = 12 in the compound interest formula, because there are 12 months in a year.

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