The sequence follows the pattern for the equation:

Therefore,

The first term for the sequence is where . Thus,

So the first term is 4.  Repeat the same thing for the second , third , fourth , and fifth  terms.

We see that the first five terms in the sequence are

To go from to , we're adding 1 to the denominator. In words, we're flipping , adding 1, then flipping it again. For example, to get from  to  we would have to flip to be 4, add 1 to get 5, then flip again to get .

The formula that shows this is

The problem isn't asking us to add the first 1,000 square numbers, but all the square numbers from 1 to 1,000. To figure out this sum, you might need to look at a list of square numbers, or play around with large squares to find the largest one under 1,000. This ends up being 31: while , which is not between 1 and 1,000. So what we're adding is the first 31 square numbers.

This means we can plug 31 in for n in that formula:

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.