FINDING THE nth TERM IN A SEQUENCE
Formula's for the nth term
Here I intend to look at what is known as the n^{th} term sequence. In doing so we will look at some examples and find the n^{th }term which is known in mathematical terms by a "u" followed by a subscript(small) _{"}_{n}_{"}_{}.
So what exactly is the connection( the pattern) between the "n" and the "u_{n}".
If we take a closer look at the u_{n }_{}we can see that it goes up in two's (2's).
(Position) U 1 2 3 4 5
( Term) U_{n }3 5 7 9 11
n x 2 2 4 6 8 10
3 + 2 = 5
5 + 2 = 7
7 + 2 = 9
9 + 2 = 11
Therefore we can write down that u_{n }= 2n, in other words we are doubling the number.
Looking again we may need to adjust that 2n slightly;
2n means 2 x n therefore,
2 x 1 = 2
2 x 2 = 4
2 x 3 = 6
2 x 4 = 8
2 x 5 = 10
These are the numbers we have got after multiplyingn x 2, you will have noticed that they are 1 less in each case than the numbers in the u_{n}_{}bracket, therefore we need to add on 1 in the formula.
Our formula now looks like this u_{n }= 2n + 1 (where the u_{n }is the n^{th }term)
2 x n = ? + 1 = u_{n}
2 x 1 = 2 + 1 = 3
2 x 2 = 4 + 1 = 5
2 x 3 = 6 + 1 = 7
2 x 4 = 8 + 1 = 9
2 x 5 = 10 + 1 = 11
Answer: u_{n }= 2n + 1
This is the type of question for the n^{th }^{}term, could be asked in an exam paper;
Find the n^{th} term for the sequence; 2, 6, 10, 14, 18 ... (etc)?
To begin with we need to change this sequence into a table!
The "n" is the position and the “u_{n}” is the term.
The position simply represents the number of digits in a sequence, in the above there are "5" [ 2(which is 1), 6(which is 2), 10(which is 3), 14(which is 4), 18( finally this is number 5 )]
(Position) U 1 2 3 4 5
_{}
( Term) U_{n }2 6 10 14 18
n x 4 4 8 12 16 20
So what exactly is the connection (the pattern ) between the "n" and the "u_{n}", in this question?
If we take a closer look at the "u_{n}" we can see that it is going up in four's (4's).
2 + 4 = 6
6 + 4 = 10
10 + 4 = 14
14 + 4 = 18
Therefore we can write down that u_{n }= 4n (which means 4 x n)
4 x 1 = 4
4 x 2 = 8
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20
These are the numbers we have got after multiplying n x 4, you will have noticed that there are 2 more in each case than the numbers in the u_{n }term, therefore we need to subtract 2 in the formula.
Our formula now looks like this; u_{n }= 4n  2 (where the u_{n}_{}is the n^{th }term)
4 x n = ?  2 = u_{n }
4 x 1 = 4 – 2 = 2
4 x 2 = 8 – 2 = 6
4 x 3 = 12– 2 = 10
4 x 4 = 16– 2 = 14
4 x 5 = 20– 2 = 18
Answer: u_{n }= 4n  2
In this example we need to find the n^{th }term for this sequence;
Find the n^{th} term for the sequence; 3, 6, 11, 18, 27 ... (etc)?
To begin with we need to change this sequence into a table!
This example is quite similar to the one above!!!!
LOOK CAREFULLY:
The "n" is the position and the “u_{n}“ is the term
Again there are five numbers and they are holding the first five POSITIONS
The position simply represents the number of digits in a sequence, in the above there are "5" [ 3(which is 1), 6(which is 2), 11(which is 3), 18(which is 4), 27( finally this is number 5 )]
(Position) U 1 2 3 4 5
( Term) U_{n }3 6 11 18 27
n^{2}=(n x n) 1 4 9 16 25
So what exactly is the connection (the pattern ) between the "n" and the "u_{n}", in this question?
Is there a set pattern?
Are they going up in 2's or 4's?  NO they are not!
3 + 3 = 6
6 + 5 = 11
11 + 7 = 18
18 + 9 = 27
In a case like this we would square (n^{2}) the numbers and see what that brings us too.
Therefore we can write down that u_{n }= n^{2} (which means n x n)
1 x 1 = 1
2 x 2 = 4
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
These are the numbers we have got after multiplying n x n, you will have noticed that they are 2 less in each case than the numbers in the u_{n }term, therefore we need to add 2 in the formula.
Our formula now looks like this; u_{n }= n^{2} + 2 ( where the u_{n }is the n^{th }term)
n x n = ? + 2 = u_{n}
1 x 1 = 1 + 2 = 3
2 x 2 = 4 + 2 = 6
3 x 3 = 9 + 2 = 11
4 x 4 = 16 + 2 = 18
5 x 5 = 25 + 2 = 27
Answer: u_{n }= n^{2} + 2
Another favourite question in exams is as follows;
The n^{th }term of a sequence is represented by this formula: u_{n }= 3n + 2
We are asked to;
(i) Find the first 4 terms
(ii) To find the 49^{th} term
( i ) Find the first 4 terms
Our formula is: 3n + 2 = u_{n }
Therefore our 1^{st} term is u_{1} = 3 x 1 = 3 + 2 = 5
2^{nd} term u_{2 }= 3 x 2 = 6 + 2 = 8
3^{rd} term u_{3 }= 3 x 3 = 9 + 2 = 11
4^{th} term u_{4 }= 3 x 4 = 12 + 2 = 14
Our first four terms are; u_{1} , u_{2} , u_{3 }, and u_{4 }
They end up as 5, 8, 11, and 14
\ First 4 terms are: 5, 8, 11, 14. (Answer:)
( ii ) We have been asked to find the 49^{th} term.
49^{th} term: This will be known as u_{49}
As before in part ( i ) the formula is; u_{n }= 3n + 2
Therefore we will replace the “_{n}” with the 49
u_{n }= 3n + 2
u_{49 }= (3 x 49) = 147 + 2 = 149
(Answer) u_{49 }= 149
Comments
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\ / \ / \ /
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Greetings ;)
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Hi , You're smart , but all comments don't help me .
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I want Math Equations .
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