Sequences and Series

Can you continue this sequence?!
2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, ….

You will need to think outside the box for this one!

To identify an integer sequence try the Online Encyclopedia of Integer Sequences.

Simply type in your sequence and choose Search.

Note that you can list, graph or even listen to the sequence!

There are further links at the foot of the page, try Puzzles for some unusual sequences which includes the sequence above.

You can also use WolframAlpha, simply type in your the sequence, scroll down the page for this example and you will see a possible sequence identification.

Typing Sequences into WolframAlpha shows several possibilities for queries on sequences.

Several of the resources mentioned on the notes page include examples of sequences, see for example the Math Centre resources on Sequences and Series. Many students find the ‘Quick Reference’ guides useful. Tony Hobson’s ‘Just the Maths’ – section 2.1 is on Series, this includes clear notes and examples and exercises. Younger students will find Craig Barton’s notes useful or the Interactive exercises from the Centre for Innovation in Mathematics Teaching (see Unit 10).

For more on WolframAlpha see the WolframAlpha pages.

By Colleen Young Posted in Number

Happy 2012!

Happy 2012! The above image (click on it) is from Jesse Vig’s geoGreeting site where you can enter a message and obtain a link which you could send in an email. It is also possible to send as an E-card. Jesse Vig noticed whilst working on a Google Maps project that a number of buildings looked like letters of the alphabet when viewed from above. and his website was born!

So to complete the new year greeting we need of course some number properties of 2012.
We could use Tanya Khovanova’s site: Number Gossip where we learn that 2012 is evil!

We’ll come back to the meaning of evil shortly! To take a look at the other properties, you will find that you can click on each for a definition.

Apocalyptic power:  a number n is called an apocalyptic power if 2n contains the consecutive digits 666 (in decimal). If we check the value of 2 to the power of 2012 – we can use WolframAlpha – we see that there is a string of three consecutive 6s.

2012 is a composite number because it is a number greater than 1 which is not prime.

2012 is deficient because the sum of all its positive divisors except itself is less than 2012.
1+2+4+503+1006=1516.

2012 is clearly even.

To understand the definition of evil requires a knowledge of binary, an evil number has an even number of 1s in its binary expansion. 2012 as a binary number is 11111011100 which we see has an even number of 1s (8). If you are interested in finding out more about binary, have a look at Math is fun on the subject.

Finally 2012 is an Ulam number. An Ulam number is a member of an integer sequence devised by and named after Stanislaw Ulam. The standard Ulam sequence begins 1, 2 … then subsequent numbers in the sequence are found by adding up two earlier ulam numbers. The number must be as small as possible and be the sum of two different earlier terms; but it can only be found in one way.
So if we start 1, 2  the next number is clearly 3, next we can have 4 because there is only one way of forming 4 from 2 earlier numbers: 1+3 (2+2 does not count as the two numbers must be different). We now have 1, 2, 3, 4… we cannot have 5 because that can be formed in two ways 1+4 or 2+3. The next in the sequence is 6 (2+4). The sequence carries on as follows: 1, 2, 3, 4, 6, 8, 11, 13, 16….. and if we carry on then 2012 is in there!

WolframAlpha can of course supply some number properties of 2012 or provide a calendar for the year …or even send us best wishes for the new year! Wishing all of you a great 2012!

By Colleen Young Posted in Number

A variety of online resources…

As you will see from the many links on this blog there are many high quality Mathematics resources available on the Internet. The collection here was inspired by a recent test taken by my Year 10 (ages 14-15) students. You can use these to explore many topics you study.

Definitions
There are many excellent reference materials online, see the reference page on this blog.
You could look up expression, equation and formula in Jenny Eather’s dictionary  for example to help you understand the difference between these terms. On the subject of expressions, equations and formulae, a related term you should be familiar with is identity. You could use this glossary for teachers to look it up.

Calculations
You can obviously use a calculator but if you don’t have one to hand you could use WolframAlpha or one of the many online calculators available such as this one from Mathisfun.

Algebra
To solve a quadratic equation using the formula try this calculator from Math Warehouse.
You could also plot the graph, note the roots where the curve intersects the x axis, ie where y=0.

WolframAlpha can always be used to check algebra, for example you may wish to check the solution to a pair of simultaneous equations.

Trial and Improvement is a method for solving equations that you cannot solve exactly.
The spreadsheet illustrations below show the solution of x3−x = 50.

The spreadsheet used here is of the extensive collection on  Mike Hadden’s MathsFiles site (Trial and Error1dp).
On the subject of Excel, you could use it to plot points and draw a graph; you could then fit a trend line as shown in the example below. Examining the numbers here you should recognise the powers of 2, we have  y = 1000×2x.

Geometry

Mathisfun has many useful Geometry resources for example you could explore quadrilaterals with this interactive and this area calculation tool allows you to check the area of several common shapes.

Trigonometry – you can use this right-angled triangle calculator by Joe Barta to check any trigonometric calculations. Simply enter the known values and state the accuracy required.

To calculate the area of a triangle if you know two sides and the included angle you could use this WolframAlpha widget.
 

Try this widget for the cosine rule.

Communicating Mathematics online can be tricky! An online whiteboard can be the answer, I have used Scriblink here:

(For other online whiteboard resources see this post)