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Sum of Consecutive Integers

How many integers from 10 to 40 can be written as the sum of 2 consecutive integers?
How many integers from 10 to 40 can be written as the sum of 3 consecutive integers?
How many integers from 10 to 40 can be written as the sum of 4 consecutive integers?
Can you find a pattern? Explain.


Problem setup

To answer these questions, I began by figuring out the first number in the sequence.  For question 1, the jump off point is 9 – but, 9 is not between 10 and 40, so I must eventually throw this out.  Next, I continued from that point.  Look at the illustration below:

I noticed that the numbers are going up by 2.  Therefore, with each multiple of 10, there will be 5 numbers that adhere to this rule.  For this question, I deduced that there are 15 numbers that are the sum of two consecutive integers.

 

Now for the next question. 

As you can see, my jump off point is 9 again; however, this time, the numbers go up in increments of 3, and are multiples of 3!  How fun!  By continuing this pattern, I found that there are 10 numbers that adhere to the rule, AND are between 10 and 40.

Third question.

Now this is a piece of cake!  Notice that the numbers increase by four with 10 as our starting point.  There are 8 numbers that adhere to our rule.

 

Plans to Solve/Investigate the Problem

 

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