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

Authors Avatar

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.

 8+9=17

2n-1

2x9-1=17

I will also use a sum I have not previously worked out.

65+66=131

2n-1

2x66-1=131

The formula for this is

3n

(Here a pattern of +3 is visible)

As before I shall now attempt to prove this formula.

9+10+11=30

3n

3x10=30

25+26+27=78

3n

3x26=78

Prediction

Although possibly premature, as it is common practice to use at least three terms in a sequence, I believe there to be a clear pattern to the formulae.  I predict that the next formula shall be 4n+1.  My reason for this is that so far n has increased by 1 and the variable has also increased by 1.  I shall test this theory for four and five consecutive numbers.  By then I also believe I shall have enough data to formulate a generic equation, which could be used to discover the equation for any number of consecutives at any term.

Join now!

        Further more, I clear pattern that the answers increase by the same number, as the number of consecutives in the sum is also visible.

The formula for this is

4n+2

(A pattern of + 4)

My initial theory was incorrect, however this has not come as a surprise, as, as I stated, it is common practice to wait for three terms before a fourth or fifth can be predicted.  Now that I have a third term I am going to submit a new theory.  I now believe that since n has increased by 1 still, but the ...

This is a preview of the whole essay