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Nine in a Line

Stage: 3 Short Challenge Level: Challenge Level:1

If n is the smallest number, the nine consecutive numbers can be expressed as:

n, n+1, n+2, n+3, n+4, n+5, n+6, n+7, n+8

We are told that they all add up to 2007

so n + n+1 + n+2 + n+3 + n+4 + n+5 + n+6 + n+7 + n+8 = 2007

so 9n + 36 = 2007

9n = 1971

n = 219

If the smallest number is 219, the largest will be 219 + 8 = 227



Alternatively, the average of the 9 numbers is 223, so they are 219, 220,..., 226, 227.

So the last number is 227.


 

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

 

 

This problem is taken from the UKMT Mathematical Challenges.
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