4. The set of natural odd numbers is divided into subsets as follows: $\{1\},\{3,5\},\{7,9,11\},\{13,15,17,19\}$,

(a) Find the first number in the 2014th subset.

(b) Is there a subset of this type that starts with 2013? Justify your answer.
## Solution.

(a) $A_{1}=\{1\}$. $A_{2}=\{3,5\}$ the sum of the elements of the set is $3+5=8=2^{3}$.

$A_{3}=\{7,9,11\}$ the sum of the elements of the set is $3^{3}, A_{4}=\{13,15,17,19\}$ the sum of the elements of the set is $4^{3}$. (2p)

The first number of the set $A_{1}$ is $0 \cdot 1+1$, the first number of the set $A_{2}$ is $1 \cdot 2+1=3$, the first number of the set $A_{3}$ is $2 \cdot 3+1=7$, and the first number of the set $A_{2014}$ is $2013 \cdot 2014+1$. (2p)

(b) The first number of the set $A_{n}$ is $(n-1) \cdot n=2 \cdot 2 \cdot 503=4 \cdot 503$ and cannot be written as a product of consecutive numbers.