Problem 3. The set of non-zero natural numbers is divided into subsets as follows:

$$
\{1,2\},\{3,4,5\},\{6,7,8,9\}, \ldots
$$

a) Find the smallest element in the 100th subset.

b) Is 2015 the largest element of one of these subsets?

Mathematical Gazette
Problem 3. The set of non-zero natural numbers is divided into subsets as follows:

$$
\{1,2\},\{3,4,5\},\{6,7,8,9\}, \ldots
$$

a) Find the smallest element in the 100th subset.

b) Is 2015 the largest element of one of these subsets?

Mathematical Gazette

Solution

a) The first 99 subsets contain $2+3+\ldots+100=5049$ elements

The first 99 subsets contain the numbers from 1 to 5049, so the smallest element of the 100th subset is 5050
b) 2015 is the largest element of the $n$-th subset if $2+3+\ldots+(n+1)=2015$.. 1p

Adding 1 to both sides results in $(n+1)(n+2)=2 \cdot 2016=4032 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 2$ 2p

Since $63 \cdot 64=4032$, it follows that 2015 is the largest element of the 62nd subset $\ldots \ldots$. $\mathbf{p}$