Okay, let's tackle this arithmetic sequence problem. So, the question is: If the fourth term of an arithmetic sequence is 200 and the eighth term is 500, what is the sixth term? Hmm, arithmetic sequences... right. Let me recall what I know about them.

First, an arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is a₁ and the common difference is d, then the nth term can be expressed as aₙ = a₁ + (n - 1)d. That formula should be helpful here.

The problem gives me two terms: the fourth term (a₄) is 200, and the eighth term (a₈) is 500. I need to find the sixth term (a₆). Let me write down the equations based on the formula.

For the fourth term:
a₄ = a₁ + (4 - 1)d = a₁ + 3d = 200.

For the eighth term:
a₈ = a₁ + (8 - 1)d = a₁ + 7d = 500.

So, now I have two equations:
1. a₁ + 3d = 200
2. a₁ + 7d = 500

I can solve these two equations simultaneously to find both a₁ and d. Once I have those, I can find a₆ = a₁ + 5d.

Let me subtract the first equation from the second to eliminate a₁. That should give me the difference in terms of d.

Subtracting equation 1 from equation 2:
(a₁ + 7d) - (a₁ + 3d) = 500 - 200
Simplify left side: a₁ - a₁ + 7d - 3d = 4d
Right side: 300
So, 4d = 300. Then, d = 300 / 4 = 75.

Okay, so the common difference d is 75. Now, plug this back into one of the original equations to find a₁. Let's use equation 1:

a₁ + 3d = 200
a₁ + 3*75 = 200
a₁ + 225 = 200
a₁ = 200 - 225 = -25.

So the first term is -25 and the common difference is 75. Now, we need to find the sixth term, a₆.

Using the formula again:
a₆ = a₁ + 5d = -25 + 5*75.

Let me compute that. 5*75 is 375. Then, -25 + 375 = 350.

So, the sixth term should be 350. Let me verify this to make sure I didn't make a mistake.

Check the fourth term: a₁ + 3d = -25 + 225 = 200. Correct.
Check the eighth term: a₁ + 7d = -25 + 525 = 500. Correct.
Check the sixth term: a₁ + 5d = -25 + 375 = 350. Yes, that seems right.

Alternatively, since the terms are in an arithmetic sequence, the difference between consecutive terms is constant. The fourth term is 200, the sixth term is two steps away (fifth and sixth), each step adding 75. So, 200 + 75 + 75 = 350. Same result.

Another way: The eighth term is 500, which is four steps after the fourth term. The difference over four steps is 500 - 200 = 300. So, each step is 300 / 4 = 75. Then, from the fourth term (200), moving two steps (to sixth term) is 200 + 75*2 = 200 + 150 = 350. Perfect, same answer.

So, all methods confirm that the sixth term is 350. I think that's solid. No mistakes detected in the calculations. The answer should be 350.
Therefore, the sixth term of the arithmetic sequence is $\boxed{350}$.
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