If I choose the consecutive numbers 2,3 and 4 then add them together, I get a total of 9. Then if I take the following set of three consecutive numbers and add them together, I get 18. And as I continue that sequence of adding together the next set of consecutive numbers, I get a total of 27 then 36 and then 45.
We can spot a pattern here that each time, the totals are increasing by 9. We can relate this to the fact that 3^2 = 9
To see if there is a pattern with more consecutive numbers, I tried the same method but with four numbers instead of three. So 2+3+4+5 = 14, next following the same pattern: We do 6+7+8+9 = 30 and then 10+11+12+13 = 46 so we can again see a pattern here of a difference of 16 each time. We can also relate this to the fact: 4^2 =16
Using this information we can predict that the next two totals will be 62 and 78.
And finally we can try this with five consecutive numbers. Now that we have spotted the pattern we can see expect there to be a difference of 25 between each of the totals. To prove it: 2+3+4+5+6 = 20 and 7+8+9+10+11 = 45 and 12+13+14+15+16 = 70.
We can see a difference of 25 between each of the totals proving that the rule would be that if you take the number of consecutive numbers and square it then you would that difference between each of the totals of the following consecutive sequence.
Using this we can predict that with six numbers, the difference would be 36 and with 7 numbers the difference would be 49.