Consecutive Integers

Some things change, some things stay the same. This is often the case in mathematics. Today, we will look at an example of when things stay the same. Consider the following two sequences of whole numbers:

4, 5, 6, 7

7, 8, 9, 10

Here are some observations of the sequences.

  •  Both sequences are 4 numbers long
  • Both sequences are consecutive; each term is one more than the previous
  • The sequences have 1 number in common (7)
  • The first sequence adds to 4+5+6+7 = 22
  • The second sequence adds to 7+8+9+10 = 34
  • The sum of the second sequence minus the sum of the first sequence is 34-22=12

Now let’s use the first three observations as conditions for another pair of sequences. How about:

11, 12, 13, 14

14, 15, 16, 17

  • In this case, the sum of the first sequence is 11+12+13+14 = 50
  • The sum of the second sequence is 14+15+16+17 = 62
  • The sum of the second sequence minus the sum of the first sequence is 62-50 = 12!

We got 12 again! Cool!

Why don’t you try with another pair of sequences, chances are you will get 12. In fact, there is no chance involved. Here is an explanation of why the difference is always 12. Look at the 14 from the second sequence and the 11 from the first sequence. Their difference is 3. The 15 and the 12 also have a difference of 3. The same goes for the 16 to 13 and 17 to 14. Hence, we have 4 pairs with a difference of 3. 4*3 = 12

Most people would stop here, having a satisfying answer to the original investigation. However, a mathematician might wonder, what happens if the sequences are longer? Or, what if they overlap more? For example:

5, 6, 7, 8, 9, 10, 11

9, 10, 11, 12, 13, 14, 15

Here the difference will be 28 (check it!). You can get the difference by the brute force method of adding up both sequences, but let’s use the same line of thinking as before to save time.

There are 7 pairs of numbers, each with a difference of 4. Hence, the answer should be 7*4 = 28. Now why do they have a difference of 4? Clearly, the sequences overlap by 3 numbers, but how can we get 4 from 3? I think it is helpful to consider the case where the sequences do not overlap at all. Here are two new sequences that do not overlap.

5, 6, 7, 8, 9, 10, 11

12, 13, 14, 15, 16, 17, 18

Either by adding up both sequences, or by using our trick, it should be obvious that the difference is 7*7 = 49. The first 7 comes from the fact that there are 7 pairs of numbers. The second 7 come from the difference of 7. The 12 is 7 larger than the 5, because the first sequence was 7 long and the second sequence didn’t overlap at all.

This suggests that each overlap will reduce the difference between the pairs. Spiraling back to our previous example, we had 4 for the difference. Observe that 7-3 = 4. 7 from the length, and 3 from the overlap.

Huzzah! A general conjecture. The difference between any two consecutive sequences of numbers is the length of the sequence times the length of the sequence minus the overlap.

7*(73)

Or in symbols:

  • Let n = the length of the sequence
  • Let k = the number of overlap
  • D = n*(n-k)

Try the conjecture on the following two sequences:

10, 11, 12, 13, 14, 15, 16, 17, 18

15, 16, 17, 18, 19, 20, 21, 22, 23

D = 9*(9-4) = 9*5 = 45

 

I have one last idea I will leave with you. What if the sequences increased at a rate other than 1? Say:

4, 6, 8, 10

10, 12, 14, 16

What would the general result be then? And how would you go about proving all of this?

 

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