To find the sum of all the integers in between 17 and 8,043, first we will find the sum of every integer from 8,043, and then we will subtract out the sum of the numbers 1-16, since those aren't between 17 and 8,043.

The sum of the first 8,043 integers is

The sum of the integers 1-16 is

Subtracting gives us

To calculate this sum, first we will need to find the sum of the positive integers, then the negative interers, then add them together.

To find the sum of the positive integers, use the formula with :

To find the sum of the negative integers, we can use the same formula as the positive numbers and then just make that answer negative.

so the negative numbers add up to .

The final answer is

 is equal to the sum of the expressions formed by substituting 1, 2, 3, 4, and 5, in turn, for  in the expression . This is simply the sum of the reciprocals of these 5 integers, which is equal to 

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:

The first difference: 

The second difference:  

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:

The first ratio: 

The second ratio: 

The sequence cannot be geometric.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:

The sequence cannot be geometric.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one also varies from term to term:

The sequence cannot be geometric.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratios of each term to the previous one is not constant from term to term:

The sequence cannot be geometric.

When evaluating a factorial, you multiply the original number by each integer less than it, stopping at 1.

For this problem, this means that

 .

Then adding 3, we get the answer 123

The value of  is defined as .

To find , simply cube the decimal number.

The closest value of this number is:  

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

Alternative Solution