Find how you can do geometric series and progression, arithmetic progression and series in this document.It has formula and easy to understand series.There are questions for practice after each content to enable you understand more.
Consider a scientist doing an experiment, he is collecting data, let us say, every day. So put x1 to
be the data collected in the first day, x 2 to be the data collected in the second day and so on, and
xn is the data collected after n days. Clearly, we are generating a set of numbers with a very
special characteristic; there is an order on the number that is we naturally have the first number,
the second number and so on. A sequence is by definition a set of all real numbers or finite set
of real numbers with the natural order given as xn n1 .
Range
Consider the sequence xn n1 .The set x1 , x2 , x3 ... xn : n 1,2,3,..., is called the range of the
sequence. In the range, there is no order.
The sequence function f (n) is the operator that generates the n th term of the sequence. In
general, f (n) xn . Where xn is the general term of the sequence.
Example
If n N in both cases (a) and (b), determine the range for the following sequences,
a) n
2
1n 4
2
b)
3n 1 4n8
Solutions
a) f (n) n 2 , 1 n 4 , that is n 1,2,3 and 4
f (1) 12 1 First term.
f (2) 2 4 2
Second term.
f (3) 3 9 2
Third term.
f (4) 4 16 2
Fourth term.
Therefore n 2 1n4 1,4,9,16
2
b) f ( n) , 4 n 8 , that is n 5,6,7 and 8
3n 1
, 2 1
f (5) First term.
3(5) 1 7
2 2
f (6) Second term.
3(6) 1 17
2 1
f (7) Third term.
3(7) 1 10
2 2
f (8) Fourth term.
3(8) 1 23
2 1 2 1 2
Therefore, , , ,
3n 1 4n8 7 17 10 23
If the general term is not given, then it may be impossible to determine it uniquely even if
many of the initial terms are provided. However by careful observation (i.e by inspection
and trial and error) a possible general term may be identified.
Example
Express the following sequences in the form xn n1
i) 7,11,15,19
4 7 10 13
ii) , , ,
1 7 17 31
Solutions
i) The difference between two consecutive terms, d 4 .
Let
S (7,11,15,19,...)
S 3 (4,8,12,16,...)
s 3
(1,2,3,4,...) nn1
4
s 3
nn1
4
S 3 4nn1
S 4n 3n1
ii) The set of the numerators is 4,7,10,13 S N
The set of the denominators is 1,7,17,31 S D
Numerator: The common difference d 3 .
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller vincentkipchirchir479. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $20.49. You're not tied to anything after your purchase.
