Okay, let's tackle this problem: finding the sum of all multiples of 7 between 100 and 200. Hmm, where do I start? Well, first, I need to figure out which numbers between 100 and 200 are multiples of 7. Then, once I have those numbers, I can add them all together to get the sum.

Wait, how do I find the first multiple of 7 after 100? Let me think. If I divide 100 by 7, that gives me approximately 14.2857. So the next integer is 15. So 15 times 7 is 105. That should be the first multiple of 7 after 100. Let me check: 14*7 is 98, which is less than 100, so yes, 15*7 is 105. Got it.

Now, the last multiple of 7 before 200. Let's divide 200 by 7. 200 divided by 7 is about 28.571. So the integer part is 28. Therefore, 28*7 is 196. That's less than 200. 29*7 is 203, which is over 200. So the last multiple is 196. Okay.

So the multiples of 7 between 100 and 200 start at 105 and end at 196. Now, how many terms are there? Let me use the formula for the nth term of an arithmetic sequence. The nth term is given by a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, which is 7 in this case.

Here, a_1 is 105, a_n is 196. So plugging into the formula: 196 = 105 + (n-1)*7. Let's solve for n. Subtract 105 from both sides: 91 = (n-1)*7. Divide both sides by 7: 13 = n-1. So n = 14. So there are 14 terms in this sequence.

Now, the sum of an arithmetic series is given by S = n/2 * (a_1 + a_n). So here, n is 14, a_1 is 105, a_n is 196. Therefore, S = 14/2 * (105 + 196) = 7 * (301) = 2107. Wait, is that right?

Wait, let me check the arithmetic. 105 + 196 is 301. 14 divided by 2 is 7. 7 times 301. Let's compute 7*300 = 2100, plus 7*1 = 7, so total is 2107. Hmm, that seems correct. But let me verify with another method to be sure.

Alternatively, I can list out all the terms and add them manually, but that might take time. Let's see:

Starting at 105, adding 7 each time:

105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.

Let me count them: 105 is the first, then 112 (2), 119 (3), 126 (4), 133 (5), 140 (6), 147 (7), 154 (8), 161 (9), 168 (10), 175 (11), 182 (12), 189 (13), 196 (14). Yep, 14 terms. Now, adding them up step by step:

105 + 112 = 217

217 + 119 = 336

336 + 126 = 462

462 + 133 = 595

595 + 140 = 735

735 + 147 = 882

882 + 154 = 1036

1036 + 161 = 1197

1197 + 168 = 1365

1365 + 175 = 1540

1540 + 182 = 1722

1722 + 189 = 1911

1911 + 196 = 2107

Okay, that matches the previous result. So the sum is indeed 2107. Wait, but just to be thorough, let me check with another formula. The sum can also be calculated by the formula S = n * average of the first and last term. Since the average is (105 + 196)/2 = 301/2 = 150.5. Then, multiplying by the number of terms, 14: 150.5 * 14. Let's compute that. 150 * 14 = 2100, 0.5*14=7, so total 2107. Yep, same result.

Therefore, I'm confident the answer is 2107.
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