ALGEBRA

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_nbracket, 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_nis 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

n²=(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 (n²) the numbers and see what that brings us too.

Therefore we can write down that u_n = n² (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 = n² + 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 = n² + 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 u₁ = 3 x 1 = 3 + 2 =   5

                             2^nd term u₂ = 3 x 2 = 6 + 2 =   8

                             3^rd term u₃ = 3 x 3 = 9 + 2 =  11

                            4^th term u₄ = 3 x 4 = 12 + 2 = 14

Our first four terms are; u₁ , u₂ , u₃ , and u₄

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 u₄₉

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

u₄₉ = (3 x 49) = 147 + 2 = 149

(Answer) u₄₉ = 149