CourseworkHelp

Triangular Square Numbers

^{Triangular Square Numbers}

Aim

The aim of this investigation is to investigate numbers that are both square and triangular. Part of this investigation will be to try to find a formula to calculate the nth triangular square number.

Initial Analysis

Triangular numbers are numbers calculated using the equation:

Triangular square numbers must satisfy the above equation, and must also be square, so:

Obtaining Data

To investigate these numbers, some examples must be discovered. Finding these numbers by hand was tried, but it was found that triangular square numbers do not occur frequently in the set of natural numbers.

A computer program was written and ran to find these numbers quicker, and therefore be more efficient. This program is shown in Appendix A, and the output file in Appendix B. This program finds all the triangular square numbers between 1 and
1 x 10¹⁶ – 1. It puts every number starting with 1 into the triangular formula, and tests if its square root is a whole number. If it is, the number produced by the triangular formula is triangular square.

This is the largest range that the programming language in which the program is written can handle with complete accuracy. The language is Gauss, which is actually designed for complex Economical calculations, but was chosen for this task because it handles numbers well. A more appropriate language to write this program in would be Maple, but a copy of the necessary program was not available.

Data Found

Note:          The original triangular number is the number that is entered into the triangular formula to produce the triangular square number.

Analysing Data

The square roots of the triangular square numbers were analysed because they are smaller and so easier to handle. If a formula is found for the nth term in the sequence of square roots, this can then be squared to find the triangular square number.

To find a formula, the differencing method was used, as seen in Appendix C, but a common difference was not found. This suggests that the formula is not a polynomial. The formula may be exponential. To check this, the ratios between successive terms should be analysed – or in other words each number in the sequence will be divided by the previous. This is shown in the following graph, for which the data can be found in Appendix D.

This shows that the ratio tends to a limit, which is a sign of exponential growth. Exponential growth is where the growth becomes more rapid in proportion to the total size.

There is a method, which can derive a general formula from an inductive definition of a sequence that grows exponentially. This method is demonstrated by finding a formula for the nth term in the Fibonacci sequence in Appendix E.

Finding a general formula

Finding an inductive definition

To find the general formula, an inductive definition is required. A mathematical method of working out an inductive definition was not known, so trial and error was used along with some educated guesswork.

After “messing” around with the numbers, it was noticed that the factor that each term must be multiplied by is about 6. It was then noted that, when multiplied by 6, the number was too big to be the correct value. By experimentation, it was discovered that subtracting the term two terms before produced the correct answer. This can be easily shown in a table:

This can then be use to produce an inductive definition:

This can then be used to create a general formula, using the method that is explained in greater detail in Appendix E.

General Form of the equation:

Finding l₁ and l₂

Replace U_n with Alⁿ.

Calculate solutions using quadratic formula.

Finding A and B

Values of for l₁ and l₂ substituted into the general formula:

A, B, l₁ and l₂ are then be entered into the general formula to find the final formula:

This is then checked on a calculator for n=0, 1, 2, 3, 4, 5, and 6. It produces the correct answer. However, they may be very small rounding errors on a calculator, so at least one has to be checked by hand. When n=2 will be checked because it was not one of the simultaneous equations used to calculate the formula.

This is the correct answer. This formula is almost totally proved. However, it is based on the inductive formula, which at the moment isn’t proved. Until it is proved the formula isn’t proved.

The formula above is the formula for the square roots of the triangular square numbers. This has to simply be squared to achieve the desired general formula:

Finding a formula using a different sequence

A general formula can also be found using the original triangular sequence. Once found, this formula must be substituted into the triangular formula to produce the required formula.

This method is only briefly described as most of it has already been shown in Appendix E or the previous section.

Finding an Inductive definition

After experimenting with the inductive definition for the square root sequence, it was found that the inductive definition of the original triangular sequence is very similar. This inductive definition has a constant of 2 added onto the end, and has different initial values:

The constant means that in this case, a particular solution must be found. This is made up of a non-specific solution, and a constant. The general form for the specific solution is:

Finding C

To work out C for the particular solution, let U_n = C " n (in the inductive definition) and manipulate the equation.

The general form for the particular equation is therefore:

To calculate l₁ and l₂ from the inductive formula, the constant is ignored. This means that the inductive definition (ignoring the constant) is the same as for the square root sequence (apart from the initial values). Therefore l₁ and l₂ are the same as before, and do not have to be worked out again.

Finding A and B

The method to find A and B is very similar to how it was done previously, only this time the constant must be present in the simultaneous equations.

This means that the final general formula is like so:

This formula was checked on a calculator for when n = 0, 1, 2, 3, 4, and 5, and all the results proved to be correct. However, there could be very small inaccuracies rounded off by the calculator. It should be checked by hand – for n = 2.

This shows that the formula is correct, although it’s not totally proved as the inductive formula from which it was created isn’t proved.

To transform this formula into the triangular square formula, it can be substituted into the triangular formula:

Possible Extension Work

Research into triangular square numbers has shown that they may have something to do with Pell Equations. Study into this field could lead to other, simpler formulae.

The formulae discovered might be able to be simplified.

Testing to see whether the two equations are the same using algebraic manipulation would help to prove them both.

A method to prove the inductive definitions from which the general formulae are derived should be looked into.

Appendix A

Appendix B

Appendix C

Attempting to use the differencing method to determine a formula for the square roots of the triangular square numbers.

Appendix D

Table showing the ratio’s of successive terms in the square root sequence

Appendix E

This is the method for calculating a general formula for an exponential sequence using an inductive definition. This example uses this inductive definition of the Fibonacci sequence:

From this definition it is known that the exponential formula must take the form:

The reason for there being two growth factors, l₁ and l₂, is because the inductive definition requires the two previous terms in the sequence. There is no constant added onto the formula because there is no constant in the inductive definition.

Finding l₁ and l₂

From the inductive definition, replace f_n with Alⁿ.

From this l₁ and l₂ can be calculated. There are two solutions of this equation, because it is a quadratic, as can easily be seen from this manipulation of the equation:

The quadratic formula can then be used to solve the equation to find both of the growth factors.

The two growth factors are therefore:

They are kept in surd form to avoid any inaccuracies when rounding to decimal.

Finding A and B

By entering the values for l₁ and l₂, into the general form of the formula, and then using simultaneous equations (substitution method), A and B can be then be found.

A and B can then be entered into the general formula with l₁ and l₂, to find the final formula:

This method of deriving a general formula automatically proves the formula, because the inductive definition is proved (as it is itself the definition of the Fibonacci sequence). However, the general formula should still be checked, in case a mistake has been made.

This formula is checked for when n = 2:

This is the correct value for when n = 2, because 5 is the sum of the two previous values in the sequence.

The values for when n = 3, 4, 5 and 6 were also checked on a calculator. When entered into the formula they produced the correct number in the Fibonacci sequence. The above method was used in case of any small inaccuracies that the calculator would round, and the calculator was used for speed.

If you still can't find any of the coursework that you are looking for, click here to look at over 30,000 GCSE, A-Level and University Level essays on Coursework.Info.