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What are consecutive sums?

Consecutive sums is just like how the name suggests it, consecutive numbers added up that makes a sum – such as 1+2+3 = 5 or 5+6+7 = 18.

Objective:

Our objective was to find out what number had the most consecutive sums from the first 100 numbers.

Work Process:

First of all I had written down the first 15 numbers and what their consecutive sums were.
Here they are.


1 = 0+1


2 = HAS NO CONSECUTIVE SUMS. 1+1 DOES NOT COUNT!


3 = 2+1


4 = HAS NO CONSECUTIVE SUMS.


5 = 2+3


6 = 1+2+3


7 = 3+4


8 = HAS NO CONSECUTIVE SUMS.

9 = 2+3+4
= 4+5

10 = 1+2+3+4


11 = 5+6


12 = 3+4+5


13 = 6+7

14 = 2+3+4+5

15 = 1+2+3+4+5
15 = 4+5+6
15 = 7+8

2, 4, 8 has no consecutive sums and 15 has the most consecutive sums so let’s use these 4 numbers to work out our rules.

2, 4, 8 are all multiples of 2 and when 2 is to the 1, 2 and 3 it ends up with 2, 4 and 8. Does it mean that powers of 2 has no consecutive sums? Let’s check the next number in the list.


16 = NO CONSECUTIVE SUMS.

16 has no consecutive sums and 32, and 64 and the list goes on.

15 has 3 consecutive sums and it’s factors are 1,3,5,15.

Let’s check the multiples of 15 up to 100.

15 = 7+8
15 = 4+5+6
15 = 1+2+3+4+5

30 = 9+10+11
30 = 6+7+8+9
30 = 4+5+6+7+8

45 = 22+23
45 = 14+15+16
45 = 7+8+9+10+11
45 = 5+6+7+8+9+10
45 = 1+2+3+4+5+6+7+8+9

60 = 19+20+21
60 = 10+11+12+13+14
60 = 4+5+6+7+8+9+10+11


75 = 37+38
75 = 24+25+26
75 = 13+14+15+16+17
75 = 10+11+12+13+14+15

90 = 29+30+31
90 = 21+22+23+24
90 = 16+17+18+19+20
90 = 6+7+8+9+10+11+12+13+14
90 = 2+3+4+5+6+7+8+9+10+11+12+13

45 & 90 has five consecutive sums as shown and their factors are.

45 = 1,3,5,9,15,45

90 = 1,2,3,5,6,9,10,16,30,45,90

Their common factors are: 3,5,9 so let’s make a theory that numbers with these factors will have more than 4 consecutive sums.










Here are some rules:

Numbers with the factors 3, 5 and 9 will most likely have more than 5 numbers.

Numbers which are a power of 2. (EG 2, 4, 8, 16, 32.) Will not have ANY consecutive sums.

Even numbers cannot be made from two consecutive sums.

Odd numbers CANNOT be made from four consecutive sums.