Question

The set of whole numbers is partitioned into subsets with the first number in the first subset, the next two numbers in the second subset, and the next three numbers in the third subset and so on. Find in terms of n a formula for the first member of the nth subset.

NB: Start with number of whole number(s) in the first partition, second partition, etc.​

Answer 1

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The first subset contains 1 number, the second subset contains 2 numbers, the third subset contains 3 numbers, and so on. So the nth subset contains n numbers.

To find the first number in the nth subset, we need to find the sum of the first n-1 numbers. The sum of the first n-1 numbers is given by the formula:

1 + 2 + 3 + ... + (n-1) = (n-1)(n-2)/2

Therefore, the first number in the nth subset is:

(n-1)(n-2)/2+1

Simplifying this expression gives:

(n^2 - 3n + 2)/2

So the formula for the first member of the nth subset is (n^2 - 3n + 2)/2.