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help needed with gce olevel maths
- Thread starter billykhan
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First of all I will start with sequences:
1. The two main skills u need to have for this chapter are recognizing common patterns and generating nth term for a sequence.
2. You may do the following to recognize a pattern in a sequence:
i. look for common differences between the terms, they may be same, increasing or decreasing from term to term.
ii. check whether its a geometric term i.e each term is multiplied by a number to get the next term.
iii. check whether the terms are divided by a number to get the next term.
iv. Also checkout for fibiccano sequence i.e 1,1,2,3,5,8.... Each term is got by adding the previous 2 terms.
These all are the common ways of recognizing patterns. You may also come across more unique patterns but don't worry, they are based on these rules usually which u can identify with practice.
Now lets move to nth term, a general formula for a sequence.
There are several methods for working out the nth term for different sequences. I guess the most common taught to o level students is ax + b where a stands for the common difference between terms and b for the 0 term, the term before the first one in the sequence. However, it is not possible that u will get a common difference every time. For that I will soon post the method, that's an all rounder method. U can solve every sequence with that. Now for geometric terms, the nth term formula common used is a^n+1 where a is common difference. Practicing the nth term, it will become easy for you. Wait until I post my all rounder formula which if u understand, u will have no problem in this chapter.
Symmetry
It is the division of a figure into two mirror images. Rotational symmetry is the times an image needs to rotate for comming back to its original, initial position. Every figure has at least rotational symmetry of 1. Generally, the lines of symmetry equals the order of rotational symmetry and when line of symmetry = 0 then rotational symmetry =1. below r some examples of symmetry of regular polygons.
square- line - 4 order- 4
parallelogram line-0 order 1
circle- line- infinite order- infinite
NOTE: The lines of symmetry for common shapes may change if there any designs made in or around the diagram in the figure.
if u also want to know symmetry in 3d figures, do tell me.
1. The two main skills u need to have for this chapter are recognizing common patterns and generating nth term for a sequence.
2. You may do the following to recognize a pattern in a sequence:
i. look for common differences between the terms, they may be same, increasing or decreasing from term to term.
ii. check whether its a geometric term i.e each term is multiplied by a number to get the next term.
iii. check whether the terms are divided by a number to get the next term.
iv. Also checkout for fibiccano sequence i.e 1,1,2,3,5,8.... Each term is got by adding the previous 2 terms.
These all are the common ways of recognizing patterns. You may also come across more unique patterns but don't worry, they are based on these rules usually which u can identify with practice.
Now lets move to nth term, a general formula for a sequence.
There are several methods for working out the nth term for different sequences. I guess the most common taught to o level students is ax + b where a stands for the common difference between terms and b for the 0 term, the term before the first one in the sequence. However, it is not possible that u will get a common difference every time. For that I will soon post the method, that's an all rounder method. U can solve every sequence with that. Now for geometric terms, the nth term formula common used is a^n+1 where a is common difference. Practicing the nth term, it will become easy for you. Wait until I post my all rounder formula which if u understand, u will have no problem in this chapter.
Symmetry
It is the division of a figure into two mirror images. Rotational symmetry is the times an image needs to rotate for comming back to its original, initial position. Every figure has at least rotational symmetry of 1. Generally, the lines of symmetry equals the order of rotational symmetry and when line of symmetry = 0 then rotational symmetry =1. below r some examples of symmetry of regular polygons.
square- line - 4 order- 4
parallelogram line-0 order 1
circle- line- infinite order- infinite
NOTE: The lines of symmetry for common shapes may change if there any designs made in or around the diagram in the figure.
if u also want to know symmetry in 3d figures, do tell me.
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Polygons
Polygon is an enclosed figure. u need to know geometric names of polygons of different sides i.e. three- triangle, four- quadrilateral, five- pentagon and so on.
The calculations involved in polygons are their exterior and interior angles. In every polygon, the sum of interior angles= 180(n-2) where n is the number of sides. The sum of exterior angles in a regular polygon equal to 360 and in usually each exterior angle is equal.
Polygon is an enclosed figure. u need to know geometric names of polygons of different sides i.e. three- triangle, four- quadrilateral, five- pentagon and so on.
The calculations involved in polygons are their exterior and interior angles. In every polygon, the sum of interior angles= 180(n-2) where n is the number of sides. The sum of exterior angles in a regular polygon equal to 360 and in usually each exterior angle is equal.
- Messages
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- Reaction score
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- Points
- 103
apart from an+b formula for nth term, I have another long but easy mehtod which can give you the solution to any possible sequence. In igcse for most we get the sequences with difference in two generations if there is unequal difference i.e. sequence: 2, 4, 8, 14, 22....
differences: 2, 4, 6, 8, .....
final ratio/second difference: 2, 2, 2....
In such cases you may use variables in quadratic forms, as they are till second difference, to substitute the sequence terms. This is general substitution for all such sequences.
i.e. sequence: a+b+c, 4a+2b+c, 9a+3b+c, 16a+4b+c, 25a+5b+c....
differences: 3a+b, 5a+b, 7a+b, 9a+b
final ratio: 2a, 2a, 2a
Note that the coefficients depends on the quadratic term: an^2 + bn+ c, n becomes term number and a,b and c are variables.
so now once you are done with it, find the values of a,b and c like this:
2a= 2
a= 1
3a+b= 2
3(1) + b= 2 (putting value of a)
b= 2-3= -1
a+b+c= 2
1+ (-1) + c= 2
c= 2
so now putting the values in quadratic term an^2+bn+c
so the nth term is n^2-n+2 (factorise further if possible)
Once you fully understand and master the concept, there should be no prolem in number sequences.
Any question??
differences: 2, 4, 6, 8, .....
final ratio/second difference: 2, 2, 2....
In such cases you may use variables in quadratic forms, as they are till second difference, to substitute the sequence terms. This is general substitution for all such sequences.
i.e. sequence: a+b+c, 4a+2b+c, 9a+3b+c, 16a+4b+c, 25a+5b+c....
differences: 3a+b, 5a+b, 7a+b, 9a+b
final ratio: 2a, 2a, 2a
Note that the coefficients depends on the quadratic term: an^2 + bn+ c, n becomes term number and a,b and c are variables.
so now once you are done with it, find the values of a,b and c like this:
2a= 2
a= 1
3a+b= 2
3(1) + b= 2 (putting value of a)
b= 2-3= -1
a+b+c= 2
1+ (-1) + c= 2
c= 2
so now putting the values in quadratic term an^2+bn+c
so the nth term is n^2-n+2 (factorise further if possible)
Once you fully understand and master the concept, there should be no prolem in number sequences.
Any question??
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Above is the all-rounder method of sequences.
- Messages
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ur welcome
- Messages
- 830
- Reaction score
- 745
- Points
- 103
actually for 3d figures, the line of symmetry is called planes of symmetry. The planes of symmetry are the layers in the 3d shape which divide them into mirror images.
look at the image above, hopefully u will get my point. If u r still confused, google it out