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• Level: GCSE
• Subject: Maths
• Word count: 1513

Round and Round -n&amp;agrave;(+1) &amp;agrave;(&amp;cedil;2) &amp;agrave;

Extracts from this document...

Introduction

Introduction

I have been asked to investigate the equation –

n→(+1) →(÷2)

(a)      (b)

I will do this by first of all, changing the first number (a) to find out if that it has any relevance to the answer that comes out of the equation at the end.

Then I am going to change the (b) number to find out weather that has anything to do with the outcome of the final number, I will also be looking for patterns and sequences in the answers.

Investigation 1

6→(+1)→(÷2)→  =   3.5

2.25

1.625

1.3125

1.15625

1.078125

1.0390625

1.01953125

1.009765625

1.004882813

1.002441406

1.001220703

1.000610352

1.000305176

As you can see, when the sum is entered in to a graphical calculator, the numbers that come out are as above. The numbers are decending from 3.5 to 1.000305176, that is the most that I have done down to so far. I predict that the numbers will eventually stop ay the number 1 as near the end there is three 0’s and before that there were two 0’

Middle

19.98632813                19.99913195

19.99316406                     19.99956598                                            19.99658203                19.99978299

19.99829102                19.9998915

19.99914551                19.9994575

19.99957276                19.99972875

19.99978638                19.99986438

19.99989319                19.99993219

19.9999466                  19.9999661

19.9999733                  19.99998305

Lower number

6→(+1)→(÷2)→  =   3.5

2.25

1.625

1.3125

1.15625

1.078125

1.0390625

1.01953125

1.009765625

1.004882813

1.002441406

1.001220703

1.000610352

1.000305176

1.000152588

1.000076294

1.000038147

1.000019074

1.00009537

1.000047685

1.000023843

1.000011922

1.000005961

1.000029805

1.000014903

1.000007452

1.000003726

1.000001863

1.000000932

1.000000466

1.000000233

1.000000177

1.000000089.

1.000000045

1.000000023

1.000000012

1.000000006

1.000000003

1.000000002

1.000000001

1.000000001

1.000000001

As you can see, when the “a” number is high, it takes a longer time to get to the final answer then when the number is lower.

When the “a”number was +1, it took only 42 attempts to get to the closest that it would get to 1, but when the “a”number was +20, after 40 attempts, it was still not even close to becoming 20. However, my analysis that the “a” number decides how long the answer takes to find could be wrong, so I am going to carry out one more investigation to certify my prediction and give me a clearer view.

Investigation 3

6→ (+10)→(÷2)→  =     89                                    9.999969483

9.5                                   9.999984742

9.75                                 9.999992371

9.875                               9.999996186

9.9375                              9.999998093

9.96875                            9.999999047

9.984375                          9.999999524

9.9921875                        9.999999762

9.99609375                      9.999999881

9.998046875                    9.999999941

9.999023438                    9.999999971

9.999511719                     9.999999986

9.99975586                      9.999999995

9.99987793                      9.999999997

9.999938965                    9.999999999

10

It took 31 attempt to get to the final number, 10, but this shows me that the size of the number, does not decide how long it takes to find the final answer.

Investigation 4

I am now going to change some more numbers in the sequence, I will change the “n” number in the sequence to see if that has any relevance to the final number given.

18→(+1)→(÷2)→  =         9.5

5.25

3.125

2.0625

1.53125

1.265625

1.1328125

1.06640625

1.033203125

1.016601563

1.008300782

Conclusion

Now I will investigate what the “b” number has to do with the equation. I will do this by changing the number, higher and lower, looking for patterns and rhythms.

Investigation 5: Changing the “b” value

6→(+1)→(÷3)→   =   2.333333333

1.111111111

0.703703703

0.567901234

0.522633744

0.507544581

0.50251486

0.75125743

0.583752476

0.791876238

0.895938119

0.631979373

0.543993124

0.514664374

As  you can see, when the “b” number has been changed, it alters the whole answer completely. There are no patterns and  the number is not heading in a pattern that I can see.

Now that I have done all my working, I think that I have found a solution.

To see if this solution works, I am going to predict something:

I predict that 6 → (+1)→ (÷5)  = 1/4

With my equation:  +x ÷y  →    x

y-1

I predict that the number will eventually round down to 0.25              (¼).

1. 1.2                                                                                        0.44                                                                                        0.288                                                                                0.2576                                                                                0.25152                                                                                0.250304                                                                                0.2500608                                                                                0.25001216                                                                        0.250002432                                                                        0.25000486                                                                        0.250000486                                                                        0.250000097                                                                        0.250000019                                                                        0.250000003                                                                        0.250000000 = ¼

My prediction was correct. I will now see if it works on other sequences.

I predict that

6 →(+3) → (÷2) = 3

will eventually end up at the number three:

4.5

3.75

3.375

3.1875

3.09375

3.046875

3.0224375

3.01171875

3.005835375

3.002917688

3.001458844

3.000729422

3.000364711

3.000182356

3.000091178

3.000045589

I was correct, it does eventually end up at 3.

This means that my equation works.

X→ (a) →(÷b) =   a

(b-1)

e.g. 1→(+2)→(÷2)=

2

2-1 =

2

1. =

2

So, I have worked out that the final rule is +x ÷y =   x

y-1

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