Suppose bot and top are positive integers with bot $$ top. Consider the array

$a[bot],a[bot+1],$dots,$a[top].$

$($a$)$ How many elements are in this array?

The array has elements.

$($b$)$ Show that if the number of elements in the array is odd, then the quantity bot $+$ top is even.

Proof: Suppose the number of elements in the array is odd.

By definition of odd, there is an integer $k$ such that the number of elements in the array $=2k+1.$

In addition, since top is the largest index in the array and bot is the smallest index in the array, then the number of elements in the array $=$

Equate the results of steps $1$ and $2$ and solve for top. After like terms are combined, the result is top $=$

Add bot to both sides to obtain bot $+$ top $),$ which is even because it is two times an integer.

Hence, bot $+$ top is an even integer $[$as was to be shown$].$

$($c$)$ Show that if the number of elements in the array is even, then the quantity bot $+$ top is odd.

Proof: Suppose the number of elements in the array is even.

By definition of odd, there is an integer $k$ such that the number of elements in the array $=2k.$

In addition, since top is the largest index in the array and bot is the smallest index in the array, then the number of elements in the array $=$

Equate the results of steps $1$ and $2$ and solve for top. After like terms are combined, the result is top $=1$

Add bot to both sides to obtain bot $+$top$=2(,)+1,$ which is odd because it is two times an integer plus $1.$

Hence, bot $+$ top is an odd integer $[$as was to be shown$].$