- Length: 10 pages
- Subject: Education - Mathematics
- Type: Essay
- Paper: #33457409

The completion of the prevailing triangular numbers of sequence possessing three more terms follows the trends below

Sn= 1-2 3-4

Through application of the graimabica rule

The general statement pertaining to the sequence of the 1,3,6,10,15,21,28,36 which are normally computed via application of the following formula Sn = n2 + n/2

1= 1 x 2 /

= 2 x 3 /

= 4 x 5 /

An nth term will be equivalent to ? n (n + 1) / 2 and it is normally the pattern recognition.

However, there exist complicated algorithm in the place of finding the difference amidst terms and if the corresponding first difference are constant. The cases that pertain to the first term being constant will subsequently results to the linear pattern. In cases where the second difference is constant, then it forms the quadratic pattern in the form of a n2 + b n + c and thus requires the finding of the pairs of the factors that would grant every term.

The results that were found on the Triangular numbers were tabulated

Stage

Stellar number

The underlying observation pertaining to the number pattern of the prevailing polynomial kind or the diverse pattern is required. The identification of the existing order of the general term is carried out through the utilization of the difference amidst the succeeding numbers. There is the utilization of mathematical methods in deriving the general term pertaining to the sequence. Moreover, there is the utilization of the technology of the GDC in the generation of the 7th and 8th terms. The technology of the GDC was utilized in generation of the extra values within the sequence accompanied by the observation of the fundamental nth term pattern of the prevailing data. Moreover, the technology of the GDC was also utilized in comprehending the developments pertaining to the sequence and any other analysis within the expected sequence.

This can be alternatively being found via the means of supplementary graphic packages. Graphic packages aids in the finding of the required general pattern and hence supporting the application of the general term.

The general term within the prevailing stage is

This is the triangular number

The general statement representing the nth triangular number within terms of n is depicted by the formula below

Sn = n2 + n/2

Alternatively, general statement can be determined via technology that involves the utilization of the calculator. The utilized calculator is the TI-84 Plus Silver Edition

n

1

2

3

4

Sn

1

11

31

61

The prevailing computation function are utilized in computing the desired Quadratic regression via application of the values of n and Sn.

The Quadratic regression expression is depicted as y= ax2 + bx + c

Taking the values of the coefficients a= 5, b= -5 and c=1 and then substituting within the equation above we get Sn= 5n2 -5n + 1

In order to test the existing validity we utilized the value of n= 4

Sn= 5n2 -5n + 1

S4= 5 x 42 -5 x 4 + 1

S4= 5 x 16 -20 + 1

S4= 80 -20 + 1

S4= 61

Therefore, the general statement is valid.

The number of the stellar numbers within every stage up to S6 accompanied by the organization of the prevailing data and the description of the pattern

There exist six-stellar numbers such that the prevailing star possesses six vertices. The stars possess diverse stages pertaining to the S1, S2, S3…. Within S1 there exists solely a single star thus it always represents a sequence of 1+12, 1+12+24, 1+12+24+36. Moreover, ever the prevailing number +12.

The prevailing stellar numbers are tabulated and represented below

Stage

Stellar number

1

1

2

13

3

37

4

73

5

6

Students are normally required to construct the stellar shape for the prevailing next stages accompanied by the existing numbers of stellar in obtaining the 6-stellar number within the fifth and the subsequent sixth stage.

The diagrams can either be handmade or via utilization of the technology. Moreover, the prevailing communication accompanied by the observation of the existing number pattern was granted. Pertaining to the observation, the correct expression of the desired terms of this underlying sequence ought to be recognized. The expression of the 7th term is represented as

. The general expression pertaining to the 6 -- stellar shape is designated as either or p n (n-1) +1. The prevailing supplementary stellar shapes are normally based on the prevailing number of the vertices that the existing students have chosen.

Sn = 1

Sn = S (n-1) +12*(n-1), where S-0 depicts the first term which is equivalent to 1

the formula is solely applicable for the stellar numbers possessing base 6

Therefore, the general declaration for the prevailing 6-stellar number, which can also be, called the numerical figures of the existing dots at the stage Sn within the terms of n follows the steps below.

As represented above, the existing series of stars is depicted by 1 + (1+12) + (1+36) + (1+72) + (1+120) + & #8230;

Taking the required sum to be identified as S,

Then the sequence would be represented by S = 1 + 13 + 37 + 73 + 121 + - + t-n?; changing one term and positioning,

S = 0 + 1 + 13 + 37 + 73 + 121 + - -t-n?. This equation contains (n+1) terms by carrying of computation that entails subtracting the existing second term from the first

0 = 1 + 12 + 24 + 36 + 48 + - -t-n?

We get the result to be t-n? = 1 + 12 + 24 + 36 + 48 + - [summation to the prevailing n terms]

t-n? = 1 + 12{1 + 2 + 3 + 4 + - to (n-1)} = 1 + 12(n-1)(n)/2 = 1 + 6n (n-1)

t-n? = 6n2-6n + 1

Therefore, the required number of dots within any Star is given through the application of the formula 6n2-6n + 1

This can be further confirmed through plugging n = 1, 2, 3,

Summation section of the formula:

As depicted above the sum of the entire dots is normally granted through the application of the formula

S = ?(6r2-6r + 1) for (r = 1 to n)

S = 6(12 + 22 + 32 + - + n2) - 6(1 + 2 + 3 + - + n) + (1 + 1 + 1 + -)

S = 6n (n+1)(2n+1)/6-6n (n+1)/2 + n

S = n[(n+1)(2n+1) - (n+1) + 1]

Expanding and simplifying the above formula we get S = n (2n2-1).

Hence the i) Numerical figures of dots within 6 star = 6(2*62-1)

= 426

ii) Numerical numbers of dots within 4 star = 4(2*42-1)

= 124

iii) Numerical figures of dots within 7 star = 7(2*72-1)

= 679

The required general statement in form of the p and n that will generates the sequence of the p-stellar numbers for any existing value of the p at stage Sn = 'p' star = p (2p2-1)

The required P- stellar numbers in the analysis of the existing 6 stellar shape relies on the numerous utilization of the values of p which are normally observed within the 6 -- stellar shape. Accurate general pattern is either or p n (n-1) +1

Thus the sixth term is given the formula = n (6(n - 1)) + 1 = 6n2-6n + 1

The production of the general statement

The production of the desired general statement in terms of the p and n that would generates the prevailing sequence of the p-stellar numbers for any existing value of p at the stage of Sn

Considering the all the prevailing general statements for the corresponding 4-stellar,5-stellar and 6-stellar at the position of the Sn in terms of the n, they form the sequence of the 4n (n-1) + 1,5n (n-1)+ 1 and 6n (n-1)+ 1 correspondingly. It is depicted that there is the desired pattern, which are identical devoid of the prevailing first term. However, the entire first terms equate to the every existing p values correspondingly thus the desired general statement in term of the variable p and n is represented as Sn = p (n) (n-1) + 1

Testing for the validity of the general statement

The process of testing the validity of the prevailing general statement entails the utilization of the next stellar shape.

For the process of testing the validity of the underlying general statement, we utilized the value p= 7 and n= 3

Sn = p (n) (n-1) + 1

S3 = 7 (3) (3-1) + 1

S3 = 21 (2) + 1

S3= 42 + 1

S3 = 43

Being that S1 ought to be equivalent to one, we therefore utilized the prevailing general statement of seven stellar in case the…