# To investigate consecutive sums. Try to find a pattern, devise a formulae and establish which numbers cannot be made using consecutive sums.

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Introduction

Introduction The Assigned Task To investigate consecutive sums. Try to find a pattern, devise a formulae and establish which numbers cannot be made using consecutive sums. To Solve the Task I am going to approach the task systematically, by running through the first twenty sums up to 5 consecutive. I shall devise a formulae and attempt to prove it. What I am Going to Do I am going to devise formulae and work out the answer to a sum I have not already done. I shall then go on to prove it by adding the sum manually. After consecutive sums have been fully investigated, I intend to take it further by investigating into consecutive minuses, squares etc. Tables shall be used where appropriate. Essential ideas may be marked in green, formulae etc., in blue. Consecutive Sums The Following Pages The following pages shall be used for the working out of many consecutive sums and their formulae. The formula for this is 2n-1 Where n is the tern. (All ready a pattern of +2 is clearly visible) I am now going to prove this theory. ...read more.

Middle

+ 3 + 4 + 5 + 6 = 20 3 + 4 + 5 + 6 + 7 = 25 4 + 5 + 6 + 7 + 8 = 30 5 + 6 + 7 + 8 + 9 = 35 6 + 7 + 8 + 9 + 10 = 40 7 + 8 + 9 + 10 + 11 = 45 8 + 9 + 10 + 11 + 12 = 50 9 + 10 + 11 + 12 + 13 = 55 10 + 11 + 12 + 13 + 14 = 60 11 + 12 + 13 + 14 + 15 = 65 12 + 13 + 14 + 15 + 16 = 70 13 + 14 + 15 + 16 + 17 = 75 14 + 15 + 16 + 17 + 18 = 80 15 + 16 + 17 + 18 + 19 = 85 16 + 17 + 18 + 19 + 20 = 90 17 + 18 + 19 + 20 + 21 = 95 18 + 19 + 20 + 21 + 22 = 100 19 + 20 + 21 + 22 + 23 = 105 The formula for this is 5n+5 7+8+9+10+11=45 5n+5 5x8+5=45 42+43+44+45+46=220 5n+5 5x43+5=220 This time my theory was correct. ...read more.

Conclusion

All odd numbers can be made. No even numbers can be made. Three Consecutive Numbers Again the pattern here is very simple. All answers follow the three times tables. If a number is not part of the three times tables i.e. if it is not divisible by three, then it cannot be made using three consecutive numbers. Four Consecutive Numbers There are several patterns which can be defined here. No odd numbers can be made using four consecutive numbers. No number from the four times table i.e. which is divisible by four, can be made using consecutive numbers. In accordance with this, no numbers divisible by eight, twelve, or divisible by any number which is divisible by four, can be made using four consecutive numbers. Five Consecutive Numbers There is a clearly definable pattern here. Only numbers divisible by five occur. In addition, all numbers, excluding five, which are divisible by five are included. It would be interesting to see if only numbers divisible by ten appear in a stream of ten consecutive numbers. However, the first term for this which I worked out earlier would immediately disprove this, as 45 is not divisible by 10. Perhaps only numbers divisible by 45 would appear? An investigation into this must be planned and set for a later date. ...read more.

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