Hello everybody, another guide for you today, this one on sequences and series! Since this is a relatively small part of the course, it will be a bit shorter than normal, but be sure to read it anyway! BOSTES like to mix it up in this section. Remember to
register and
ask a question below, and check out the
notes for both 2U and Ext 1, which go into lots more depth and are an awesome resource if you are in need of extra revision. Right, without further ado, lets go!
First thing you should know is the difference between an arithmetic series and a geometric one. An arithmetic series is where the next term is obtained by adding a common difference (+ or -), and has the following formulas which apply:
d\\ \\ { S }_{ n }=\frac { n }{ 2 } \left[ 2a+(n-1)d \right] )
A geometric series is a series where the next term is obtained by multiplying the previous term by a common ratio, with the following formulas:
 }{ 1-r } )
The first tip, make sure you apply the correct formula!
Many questions in these sections are straightforward, such as this first example.
Example One (2014 HSC): Evaluate the arithmetic series 2 + 5 + 8 + 11 + ··· + 1094.[/b]
An easy two marks, like many arithmetic questions. A simple use of formula, they will rarely get more complex than rearranging to find a variable.
d\\ 1094=2+3(n-1)\\ 3n-3=1092\\ 3n=1095\\ n=365\\ \\ \therefore \quad { S }_{ n }=\frac { n }{ 2 } \left[ 2a+(n-1)d \right] \\ \\ \quad \quad \quad \quad =\frac { 365 }{ 2 } \left[ 4+(3\times 364) \right] \\ \\ \quad \quad \quad \quad =200020 )
Also remember the cool shortcut for finding the sum of \(n\) terms of an arithmetic series:
This is easy to prove (in fact, they could ask!).
Geometric sequence questions can get nastier, and there is no better example of that than the financial mathematics style questions. There are various types, but let's go through one from last years paper.
Example 2 (HSC 2014): At the start of a month, Jo opens a bank account and makes a deposit of $500. At the start of each subsequent month, Jo makes a deposit which is 1% more than the previous deposit. At the end of every month, the bank pays interest of 0.3% (per month) on the balance of the account.
a) Explain why the balance of the account at the end of the second month is  }^{ 2 }+$500(1.01)(1.003) )
These questions can really difficult to interpret and manage. Let's break it down step by step, which is exactly what you should do in the exam.
At the start of the first month, she deposits $500. On the last day of that month, she receives 0.3% interest, so the balance (B) is:
 )
On the first day of the second month, she adds $500 plus 1% more, so:
+$500(1.01))
At the end of that month, 0.3% interest is applied to the whole account, so:
^2 +$500(1.01)(1.003) )
Taken through step by step, this is actually not a terribly difficult proof. The hard part is when it has to be generalised, but start by doing a simple breakdown like this. Perhaps do one more to show that after three months:
^3+$500(1.01)^2(1.003)^2+500(1.01)(1.003) )
This should make the general case more obvious.
(b) Find the balance of the account at the end of the 60th month, correct to the nearest dollar. For sixty months:
^{60}+500(1.01)(1.003)^59+500(1.01)^2(1.003)^{58} \rightarrow +500(1.01)^{59}(1.003) )
Spotting that GP is tricky, but it will ALWAYS be there. Those three lines of working, which, neglecting the tricky numbers, is essentially just using a formula, was 3 marks! Super easy, once you get the hang of correctly interpreting the question.
The last piece of the puzzle is limiting sums. Arithmetic progressions all tend to infinity as they go on, but GP's may not. Those that don't have what is called a limiting sum, and it occurs when:
You could be asked a variety of things, but all of them are easy in comparison to the financial math questions. Time is better spent there, but limiting sums were worth about 6 marks in my paper, pretty sizeable! Don't forget them. The formula you need is below:
So that's about it! A short part of the course, with some of the easiest (and hardest) marks you'll receive in the exam! Be sure to
ask any questions below, or head on over to the new question forum and see if you can get some help, or even give some! Be sure to check out the notes, check out the forums, check out everything! It's an awesome community with heaps of stuff to help you guys succeed
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