Geometry in Higher Education: A Task-Oriented Perspective

A sequence also called a progression is an ordered list of numbers. Every number in the sequence is called an element. The addition of all numbers in a sequence is called a series. There are many types of sequences and series. A sequence that has an upper limit is called a finite sequence or correspondingly a finite series. The type that has no upper limit is called an infinite sequence and an infinite series if it is a series. There are more classifications such arithmetic, geometric sequences and many others. The types of series and sequence that are useful are the kinds that have a formula that can predict any term in the sequence or series.

            Of all the many types of series, two types are particularly popular they are commonly used these are, arithmetic sequences and geometric sequences. In a geometric sequence, every next term divides with the previous term to yield a constant value called a common ratio, in an arithmetic sequence on the other hand, the previous subtracts from the next to yield a constant value, the result is called a common difference (Grigorieva, 2016).

Task 1

a) To solve this equation 1- +more easily it can be rewritten in the form shown below ∑(- if  the first value of  is chosen as 53 degrees which when converted to radians yields 0.9250

Degrees are converted to radians as follows 1800= radians, what about 530

530 */1800

which equals 0.9250 radians as stated previously

b) The other two values of  are set at 43 degrees and 23 degrees, these too are converted to radians in the same way as done in the above exercise as shown table 430=0.7505 and 230= 0.4014 radians. The addition of all 13 steps using excel are shown below: that is, the equation ∑(- is used to calculate the sum of all the 13 terms as illustrated in the excel tables below.

c) As the angles are reduced from 530 then 430 then finally 230, the value at which the sum of all terms converges grows steadily to a bigger value. That is from 0.6018 for 0.9250 radians, to 0.9205 for 0.4014 radians or 230.

Task 2

The sequence shown below is a geometric sequence, a geometric sequence is a sequence which any of its elements after the first can be found by multiplying the preceding term by a constant called as the common ratio, it is usually denoted by the letter ( r). The common ratio (r) is found by dividing any term by the preceding term for example,     r =, ,....

a) If the length of the first string that is chosen as 4 meters, if it is cut into two halves then one of the haves is kept in the box,. The contents of the box will therefore look as shown below              

+++…

This is a geometric sequence; the sum of all the terms is therefore called a geometric series.

The common ratio is (1 divided by2) which is

The sum of first terms can be found using the following formula Sn=a1where r is the common ratio, Sn is the sum of n terms; a1 is the first term which in case is 1/2, adding the first 100 terms as shown below yields

            S100=2  solving this equation yields 4 meters.

b) The series above appears to converge at 4 meters, the original length of the string that was chosen

c) If the length of the string chosen in the second time round is 2 meters, the contents of the box this time will look as shown below

            1++++…

The sum of 120 terms

            S120=1= 2 meters

d) Just like the first example, the series appears to converge at the original length of the string

e) For as long as this is a geometric series this is always going to be the case. The justification is the ratio is constant, the formula does not change either, plus if you plug in a really large number say infinity, and the value 0.5∞ vanishes so ultimately, the length of the first piece of string that is put in the box is multiplied by 2.

f) The result is completely surprising, it is not expected, you would for example expect the length of all the contents in the box to add up to a length fairly less than the length we started from since some of the rope was not put in the box.

Task 3

            Iteration is a technique in mathematics where an equation is solved several times. It is where the result of a previous calculation is used as the input of the next calculation. This is repeated for as many times as possible until an accurate answer is found. Usually, a starting value is provided, this value is plugged into a formula the result is then taken as a new starting point. Iteration is a tool that is used to solve many mathematical problems. It has many applications too, for example iteration can be used to find the real roots of an equation (Kelley,1995).

a) For k = 5

Un+1=(Un+)  If  U1=1 then we can plug it in as Un , it then follows that Un+1

is our U2, K=5. The working is shown below.

U1=1 the first term

U2=(1+)=3,  U3=(3+)=2.3, U4=(2.3+)=2.237

For k = 6, we repeat the process we did above

            U2=(1+)=3.5,  U3=(3.5+)=2.607, U4=(2.607+)=2.454

For k = 7

            U2=(1+)=4,  U3=(4+)=2.875, U4=(2.875+)=2.655

For k = 8

            U2=(1+)=4.5,  U3=(4.5+)=3.138, U4=(3.138+)=2.843

b)  Using excel with different values of K

c) Iterations using different values of k and U1=4

 U1=4

For k = 17,

            Un+1=(Un+)  then iterating three times as shown below

            U1=4

            U2=(4+)=4.125,  U3=(4.125+)=4.123, U4=(4.123+)=4.123

For k = 18

            U2=(4+)=4.25,  U3=(4.25+)=4.243, U4=(4.243+)=4.243

For k = 19

            U2=(4+)=4.375,  U3=(4.375+)=4.359, U4=(4.359+)=4.359

For k = 20

            U2=(4+)=4.5,  U3=(4.5+)=4.472, U4=(4.472+)=4.472

For k = 21

            U2=(4+)=4.625,  U3=(4.625+)=4.583, U4=(4.583+)=4.5825

For k = 22

            U2=(4+)=4.75,  U3=(4.75+)=4.691, U4=(4.691+)=4.690

For k = 23

            U2=(4+)=4.875,  U3=(4.875+)=4.796, U4=(4.796+)=4.796

For k = 24

            U2=(4+)=5,  U3=(5+)=4.9, U4=(4.9+)=4.899

d) Using excel as shown below with U1=2 and K=14 then iterating 15 times, shows that the iterations converge at 2.4495.

For k = 14

            U2=(2+)=3,  U3=(3+)=2.5555, U4=(2.5555+)=2.6779

e) This function will converge at a specific number that number depends on the choice U1 and the choice of constant K

If k=16, and the firs value is set at U1=3

Un+1=(Un+), as shown below, plugging in the values of U1=3 and k=16 and iterating three times yields.

U2=(3+16/3)=2.083, U3=(2.083+16/2.083)=2.4406, U4=(2.4406+16/3)=2.2491 it will very likely converge at 2.3.

Summary and Conclusion

            Adding all the values in the series of task 1 appears to converge at the cosine of the respective angles, for example, the values of 230 or 0.4014 radians sums up to 0.9205 which is  the cosine of 0.4014 radians so do values of 430

and 530. In task two, it is curious that the total of all the pieces in the box appear to converge at the length it started from. In task 3, the changes in the results are steep in the starting iterations, as the iterations increase in number; however, the changes peter out until they totally vanish.

Bibliography

Grigorieva, E., 2016. Methods of solving sequence and series problems. Cham, Springer International Publishing. http://dx.doi.org/10.1007/978-3-319-45686-7.

Kelley, T.,1995. Iterative methods for linear and nonlinear equations. Philadelphia, Pa, Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104)