What a familiar progression!

Take for instance the number set 10, 20, 30, 40, 50, 60. There are many ways to find the sum, but since the difference between each number is the same, you can add up the opposite numbers (first with last, second with second to last, third with third to last) and they'd all have the same sum. Here it would be 70 (10 + 60), and since there are three of those like sets, you can multiply that by 3 to get the sum. In this problem here, 1 + 2017 = 2018. 2018 x 3 = 6054.

Relevant wiki: Arithmetic and Geometric Progressions Problem Solving

Since the difference between each consecutive numbers are the same, the given sequence is an arithmetic progression whose sum is given by $\frac{n}{2}(a+l),$ where $a$ is 1st term and $l$ is last term. So we do not need the middle terms at all and the answer is 6054.

Since the differences between two consecutive numbers in the sequence are all the same, the sequence is an arithmetic sequence.

First, you will start with the formula:

$a_{n}$ = $a_{i}$ + d(n - 1)

$a_{n}$ = nth term in the sequence $a_{i}$ = First term in the sequence n = number of terms in the sequence d = the common difference

Essentially, in this problem, we're looking for d .

First, we plug in what we already know about the problem.

$a_{n}$ = 2017 $a_{i}$ = 1 n = 6 d = ?

Next, we plug in all of the known information in our formula:

2017 = 1 + d(6 - 1)

Next, we solve for d:

1. 2017 = 1 + d(6-1)
2. 2017 = 1 + d(5)
3. 2017 - 1 = 1 + d(5) - 1
4. 2016/5 = d(5)/5
5. 403.2

Now that we have our common difference d , we can find the missing four terms by continually adding 403.2 onto term n-1. So we get the following sequence:

1, 404.2, 807.4, 1210.6, 1613.8, 2017.

We then find the sum of all of the terms:

1 + 404.2 + 807.4 + 1210.6 + 1613.8 + 2017 = $\boxed{6054}$

Using the same idea as the famous 1-100 sum, we take pairs which are equidistant from the middle (i.e. 1 and 2017) and add them. In this case, the sums are 1+2017=2018, so 2018*3=6054 is our final answer!

Since the differences between two consecutive numbers are all the same, the numbers form an arithmetic progression .

Now, let $a$ be the initial term and $d$ be the constant difference between two consecutive numbers.

Now, clearly:

$a = 1 \ \text{and} \ a + 5d = 2017$

And the sum of the numbers is equal to $6a + 15d = 3a + 3a + 15d = 3(a) + 3(a + 5d) = 3\cdot1 + 3\cdot2017 = \boxed{6054}$ which is the answer.