Sum of three distinct members of the set
Algebra
Level
3
How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19} ?
-Problem from the AMC 2019(American Mathematics Competition).
The answer is 13.
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1 solution
Each of the elements of the set is nothing but 3n + 1 where n = 0,1...6.
So if any three numbers from this set are added , the resultant sum will be divisible by 3.
The smallest sum in the resulting set is 12 and the largest sum is 48.
So the set of numbers in the series containing these sums will range between 12 and 48. S= {12,15,18..48). . Hence the total number of different integers which can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19} is the number of terms in the Arithmetic progression S with common difference = 3. This is equal to 13.