Unit 14: Sequences, Series, Mathematic Induction, and Statistics

mean

- the average of all data values - to calculate the mean sum all the numbers in the data set and divide the sum by the number of values

histogram

- the height of each bar corresponds to the frequency of events in that particular category - is to quantitative data what a bar chart is to categorical data, however, unlike a bar chart, both axes of a histogram have numerical scales, and the rectanguar bars on adjacent intervals have no intentional gaps between them

time plot

- the input variable is time - enables us to analyze the patterns as the variable changes over time - plots of data points in a 2-dimensional coordinate system in which the points are connected by line segments

median

- the middle number when the data are ordered from smallest to largest if there is an odd number of data values; if there is an even number of data values, then the median is the average of the two middle numbers

mode

- the most frequent data value; any data set that does not have a value repeated more than once does not have a mode

how to use mathematical induction in a proof

1.) Show that a first statement, P₁, is true. 2.) Show that if any statement is true, each successive statement is also true. If P↓k↓ is true, then P↓k + 1↓ is also true. 3.) Conclude that since P₁ is true, and P↓k↓'s truth implies P↓k + 1↓'s truth, then all statements must be true.

geometric sequence

A sequence {a↓n↓} is a geometric sequence if it can be written in the form {a, a * r, a * r², . . ., a * rⁿ⁻¹, . . .} for some nonzero constant r. The number r is called the common ratio. Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r: a↓n↓ = a↓n - 1↓ * r (for all n ≥ 2) explicit formula: a↓n↓ = a₁ * r^(n - 1)

arithmetic sequence

A sequence {a↓n↓} is an arithmetic sequence if it can be written in the form {a, a + d, a + 2d, . . ., a + (n - 1)d, . . .} for some constant d. The number d is called the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d: a↓n↓ = a↓n - 1↓ + d (for all n ≥ 2). explicit formula: a↓n↓ = a₁ + (n - 1)d

infinite series

An infinite series is an expression of the form (∑ with ∞ over sigma, a↓n↓ on the right side of sigma, and n - 1 on the bottom of sigma) = a₁ + a₂ + . . . + a↓n↓ + . . . - the sum of a sequence with an infinite number of terms

summation notation

In summation notation, the sum of the terms of the sequence {a₁, a₂, . . ., a↓n↓} is denoted ∑ with n over sigma, a↓k↓ on the right side of sigma, and k = 1 on the bottom of sigma which is read "the sum of a↓k↓ from k = 1 to n." The variable k is called the index of summation.

principle of mathematic induction

Let P↓n↓ be a statement about the integer n. Then P↓n↓ is true for all positive integers n provided the following conditions are satisfied: 1. (the anchor) P₁ is true; 2. (the inductive step) if P↓k↓ is true, then P↓k + 1↓ is true.

sum of a finite arithmetic sequence

Let {a₁, a₂, a₃, . . ., a↓n↓} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is (∑ with n over sigma, a↓k↓ on the right side of sigma, and k = 1 on the bottom of sigma) = a₁ + a₂ + . . . + a↓n↓ = n((a₁ + a↓n↓)/2) = (n/2)(2a₁ + (n - 1)d)

sum of a finite geometric sequence

Let {a₁, a₂, a₃, . . ., a↓n↓} be a finite geometric sequence with common ratio r ≠ 1. Then the sum of the terms of the sequence is (∑ with n over sigma, a↓k↓ on the right side of sigma, and k = 1 on the bottom of sigma) = a₁ + a₂ + . . . + a↓n↓ = (a₁(1 - rⁿ))/(1 - r)

limits of infinite sequences

Let {a↓n↓} be a sequence of real numbers, and consider lim a↓n↓ as n → ∞. If the limit is a finite number L, the sequence converges and L is the limit of the sequence. If the limit is infinite or nonexistent, the sequence diverges.

the Fibonacci sequence

The Fibonacci sequence can be fined recursively by a₁ = 1 a₂ = 1 a↓n↓ = a↓n - 2↓ + a↓n - 1↓ for all positive integers n ≥ 3. - the successive terms of the Fibonacci sequence are obtained by adding the 2 previous terms (1, 1, 2, 3, 5, 8, 13)

sigma notation

The capital Greek letter ∑ indicates that we are to sum the terms of the sequence created using the rule, f(k). The starting value for the sequence is given by k = 1, and n gives the ending value for the sequence.

sum of an infinite geometric series

The geometric series (∑ with ∞ over sigma, k = 1 below sigma, and a * r^(k - 1) on the right side of sigma) converges if and only if |r| < 1. If it does converge, the sum is a/(1 - r).

normal distribution

a function that represents the distribution of many random variables as a summetrical bell-shaped graph

finite sequence

a sequence whose domain is finite

infinite sequence

a sequence whose domain is infinite

sequence

an ordered progression of numbers

recursive

applying a rule or formula to its results

recursive sequence

create the next term of the sequence based on the previous terms

stemplot (or a stem-and-leaf plot)

each number in the data is split into a stem, consisting of its initial digit or digits, and a leaf, which is its final digit

categorical variable

if the variable identifies each individual as belonging to a distinct class, such as male or female

quantitative variable

if the variable takes on numerical values for the characteristic being measured

frequency table

it lists only the number of entries in specific categores

box-and-whisker plots

minimum - smalles value in the data set first quartile - median of the lower half of the data set; the lower half is defined to be the values below the median of the entire data set median - middle value of the data set; this is also the second quartile third quartile - median of the upper half of the data set; the upper half is defined to be the values above the median of the entire data set maximum - largest value in the data set range - the difference between the largest and smallest data values interquartile range - the difference between quartile one and quartile three outliers - data values which lie more then 1.5 times outside the inner quartile range - a short box compared to a long whisker length tells us most of the data is grouped together around its median - a longer box comapred to its whiskers tells us the data is quite spread out across their range

enumerative induction

one reasons from specific cases to the general principle by considering all possible cases

variable

the characteristic of the individuals being identified or measured; they are either categorical or quantitative

kth term

the kth number in a sequence

the tower of Hanoi solution

the minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2ⁿ - 1

individuals

the objects described by a set of data

the sequence of partial sums

the sequence of nth partial sums of a series

series

the sum of a sequence

finite series

the sum of a sequence with a finite number of terms

diverges

the sum of an infinite series is infinitely large

converges

the sum of an infinite series never grows beyond a specific value - occurs only if the common ratio is -1 < r < 1 - the sum of a converging infinite geometric series is defined by the equation: S = a₁/(1 - r)

partial sum

the sum of part of a sequence

back-to-back stem-plots

use the same stems, but leaves from one set of data are added to the left, while leaves from another set are added to the right

when do we reason by deduction?

when we reason from general principles to draw conclusions about specific cases

when do we reason by induction?

when we use evidence derived from particular examples to draw conclusions about general principles

standard deviation

µ = mean of set variance = σ² n = number of elements in set x↓i↓ = ith element of set