Corporate Headquarters 301 Coromar Drive Goleta, California 93117. Oxygen Therapy Users 1-800-630-3144 (toll free) Customer Support for Existing Patients. Numbers can be installed on Mac OS X 10.10 or later. This Mac app was originally produced by Apple Inc. The most popular versions of the application are 3.5, 3.2 and 2.3. 1) First, download and install the iWork ’09 trial (.dmg). 2) Launch Pages, Numbers and Keynote, then quit each app. 3) Launch the Mac App Store, and go to the Updates tab. When calculating equivalent ratios you must multiply or divide both numbers in the ratio. This keeps both numbers in direct relation to each other. So, a ratio of 2/3 has an equivalent ratio of 4/6: in this ratio calculation we simply multiplied both 2 and 3 by 2.
To find a missing number in a Sequence, first we must have a Rule
Sequence
A Sequence is a set of things (usually numbers) that are in order.
Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion.
Finding Missing Numbers
To find a missing number, first find a Rule behind the Sequence.
Sometimes we can just look at the numbers and see a pattern:
Example: 1, 4, 9, 16, ?
Answer: they are Squares (12=1, 22=4, 32=9, 42=16, ...)
Rule: xn = n2
Sequence: 1, 4, 9, 16, 25, 36, 49, ...
Did you see how we wrote that rule using 'x' and 'n' ?
xn means 'term number n', so term 3 is written x3
And we can calculate term 3 using:
x3 = 32 = 9
We can use a Rule to find any term. For example, the 25th term can be found by 'plugging in' 25 wherever n is.
x25 = 252 = 625
How about another example:
Example: 3, 5, 8, 13, 21, ?
After 3 and 5 all the rest are the sum of the two numbers before,
That is 3 + 5 = 8, 5 + 8 = 13 etc, which is part of the Fibonacci Sequence:
3, 5, 8, 13, 21, 34, 55, 89, ...
Which has this Rule:
Rule: xn = xn-1 + xn-2
Now what does xn-1 mean? It means 'the previous term' as term number n-1 is 1 less than term number n.
And xn-2 means the term before that one.
Let's try that Rule for the 6th term:
x6 = x6-1 + x6-2
x6 = x5 + x4
So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:
x6 = 21 + 13 = 34
Many Rules
One of the troubles with finding 'the next number' in a sequence is that mathematics is so powerful we can find more than one Rule that works.
What is the next number in the sequence 1, 2, 4, 7, ?
Here are three solutions (there can be more!):
Solution 1: Add 1, then add 2, 3, 4, ...
So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc...
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, ...
(That rule looks a bit complicated, but it works)
Solution 2: After 1 and 2, add the two previous numbers, plus 1:
Rule: xn = xn-1 + xn-2 + 1
Sequence: 1, 2, 4, 7, 12, 20, 33, ...
Solution 3: After 1, 2 and 4, add the three previous numbers
Rule: xn = xn-1 + xn-2 + xn-3
Sequence: 1, 2, 4, 7, 13, 24, 44, ...
So, we have three perfectly reasonable solutions, and they create totally different sequences.
Which is right? They are all right.
And there are other solutions ...
... it may be a list of the winners' numbers ... so the next number could be ... anything! |
Simplest Rule
When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.
Finding Differences
Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.
Here is a simple case:
The differences are always 2, so we can guess that '2n' is part of the answer.
Let us try 2n:
The last row shows that we are always wrong by 5, so just add 5 and we are done:
Numbers For Mac 10.12.6
Rule: xn = 2n + 5
OK, we could have worked out '2n+5' by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.
Second Differences
In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...
... and then find the differences of those (called second differences), like this:
The second differences in this case are 1.
With second differences we multiply by n22
In our case the difference is 1, so let us try just n22:
| n: | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Terms (xn): | 1 | 2 | 4 | 7 | 11 |
| n22: | 0.5 | 2 | 4.5 | 8 | 12.5 |
| Wrong by: | 0.5 | 0 | -0.5 | -1 | -1.5 |
We are close, but seem to be drifting by 0.5, so let us try: n22 − n2
Wrong by 1 now, so let us add 1:
| n22 − n2 + 1 | 1 | 2 | 4 | 7 | 11 |
|---|---|---|---|---|---|
| Wrong by: | 0 | 0 | 0 | 0 | 0 |
We did it!
The formula n22 − n2 + 1 can be simplified to n(n-1)/2 + 1
So by 'trial-and-error' we discovered a rule that works:
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...
Other Types of Sequences
Read Sequences and Series to learn about:
And there are also:
And many more!
In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.
Numbers 10123
Median Calculator Instructions
Numbers For Mac 10.12 6
This calculator computes the median from a data set:
To calculate the median from a set of values, enter the observed values in the box above. Values must be numeric and may be separated by commas, spaces or new-line. You may also copy and paste data into the text box. You do not need to specify whether the data is from a population or a sample, unless you will later examine the variance or the standard deviation. Press the 'Submit Data' button to perform the computation. To clear the calculator and enter a new data set, press 'Reset'.
What is the median
The median is a measure of central tendency. It represents the value for which 50% of observations a lower and 50% are higher. Put simply, it is the value at the center of the sorted observations.
Median formulas
This calculator uses two different formulas for calculating the median, depending on whether the number of observations is odd, or it is even:
When the number of observations is odd the formula is:
How Do I Update My Mac To 10.12 6
When the number of observations is even the formula is:
where n is the number of observations.