<Question>
What is the sum of all of the multiples of 7 between 100 and 200?
</Question>

<think>
<Plan_1>
Okay, so I need to find the sum of all the multiples of 7 between 100 and 200. Hmm, let's start by understanding what exactly the problem is asking for. Multiples of 7 are numbers like 7, 14, 21, and so on. But here, we need those multiples that lie between 100 and 200. So first, I should figure out which is the first multiple of 7 after 100 and which is the last multiple of 7 before 200.

Let me think. To find the first multiple of 7 greater than 100, I can divide 100 by 7 and then round up to the nearest whole number.

<prompt_1.1> Calculate the first multiple of 7 greater than 100 by dividing 100 by 7 and rounding up. </prompt_1.1>

Similarly, to find the last multiple of 7 less than 200, I can divide 200 by 7 and round down.
200 divided by 7 is approximately 28.57, so the last integer is 28. So 28 times 7 is 196. Let me check: 28*7=196.
</Plan_1>

<Execution_1>
<execution_1.1>
So 100 divided by 7 is approximately 14.2857. So the next integer is 15. Then, multiplying 15 by 7 gives 105. So 105 is the first multiple of 7 after 100. Wait, but let me confirm: 14*7 is 98, which is less than 100, so yes, 15*7 is 105. That makes sense.

Now, the last term before 200 would be 28*7=196, as I thought earlier. Because 29*7 is 203, which is over 200. So 196 is the last term.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
So now, the problem is to sum all the numbers from 105 to 196, inclusive, that are multiples of 7. This is an arithmetic sequence, right? Because each term increases by 7. The sequence would be 105, 112, 119, ...., 196.

To find the sum of an arithmetic sequence, I remember the formula: Sum = (n/2)*(a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. So I need to find n first.

How do I find the number of terms in the sequence? Well, since it's an arithmetic sequence with common difference d=7, the number of terms can be calculated by taking the difference between the last term and the first term, dividing by the common difference, and then adding 1 (because both endpoints are included).

<prompt_2.1> Calculate the number of terms in the arithmetic sequence using the formula: n = ((last term - first term) / common difference) + 1. </prompt_2.1>
-----

Alternatively, using the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)*d. We know that the last term is 196, so plugging in: 196 = 105 + (n - 1)*7.

<prompt_2.2> Use the formula for the nth term of an arithmetic sequence to verify the number of terms. </prompt_2.2>
-----

Alternatively, I could have used the formula for the sum of an arithmetic series: S = n/2*(a1 + an). We already found n=14, a1=105, an=196. So 14/2*(105 + 196) = 7*201=1407.

Alternatively, maybe there's another way to compute this sum without identifying a1 and an first. For example, using the formula that involves the first term, last term, and number of terms, which is what we did.
Alternatively, perhaps using the average of the first and last term multiplied by the number of terms. The average of 105 and 196 is (105 + 196)/2 = 301/2 = 150.5. Then multiplying by the number of terms, 14: 150.5*14. Let's compute that.

<prompt_2.3> Calculate the sum using the average of the first and last term multiplied by the number of terms. </prompt_2.3>

Alternatively, maybe I can use the formula for the sum of multiples of 7. Since each term is 7 times a number. So 7*(15 + 16 + 17 + ... +28). Then the sum would be 7 times the sum of numbers from 15 to 28. Let's compute that.

<prompt_2.4> Calculate the sum of multiples of 7 using the formula: 7*(sum of numbers from 15 to 28). </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's compute that.

First term (a1) is 105, last term (an) is 196. Common difference (d) is 7. So the difference between last and first terms is 196 - 105 = 91. Then, dividing by 7 gives 91/7 = 13. Then adding 1 gives 14. So there are 14 terms in this sequence.
</execution_2.1>

<execution_2.2>
Let's solve for n. Subtract 105: 196 - 105 = (n - 1)*7. 91 = (n - 1)*7. Divide both sides by 7: 13 = n - 1. So n = 14. That matches what I had before. So there are 14 terms.
</execution_2.2>

<execution_2.3>
150*14 = 2100, and 0.5*14=7, so total is 2100 +7=2107. Wait, that's not matching the previous result. Wait, that can't be right. Wait, 150.5*14. Let me do it step by step. 150*14=2100, 0.5*14=7, so 2100+7=2107. Wait, but that contradicts the previous result. So there must be a mistake here.
</execution_2.3>

<execution_2.4>
The sum from 15 to 28 can be calculated as the sum from 1 to 28 minus the sum from 1 to 14. The formula for the sum from 1 to n is n(n+1)/2.

So sum from 1 to 28 is 28*29/2 = 406. Sum from 1 to 14 is 14*15/2 = 105. Therefore, sum from 15 to 28 is 406 - 105 = 301. Then multiplying by 7 gives 301*7=2107. Wait, that's different from before. Wait, this can't be right. Wait, 301*7. Let me compute that.

300*7=2100, 1*7=7, so total is 2107. But earlier, we had 1407. That's a problem. There's a discrepancy here. So which one is correct? Hmm. Let's check where the mistake is.

Wait, if the original calculation using the arithmetic sequence gave 1407, but this method gives 2107, there's a problem. Let me check the second method again.

Wait, the multiples of 7 between 100 and 200 start at 105 (15*7) and end at 196 (28*7). So the coefficients are from 15 to 28 inclusive. So the number of terms is 28 -15 +1 =14. That's correct. Then sum from 15 to28 is sum(15:28) = sum(1:28) - sum(1:14) = (28*29)/2 - (14*15)/2 = 406 -105=301. Then 301*7=2107. But this contradicts the first method. So which is correct?

Wait, there's a mistake here. Wait, if each term is 7 times the coefficient, then the sum is 7*(sum of coefficients). But if the coefficients are 15 to28, then sum of coefficients is 301, so total sum is 301*7=2107. But this contradicts the first method where we had 14 terms, first term 105, last term 196, sum=14/2*(105+196)=7*301=2107. Wait, but earlier I thought the first term was 105 and last term 196, so n=14, so sum=14/2*(105+196)=7*301=2107. Wait, but in the first method, I thought I had 14 terms, but then when I listed them out, I only wrote 14 terms. Wait, but in the first method, when I calculated n=14, that's correct. Then sum is 14/2*(105+196)=7*301=2107. So why did I initially think it was 1407? Because maybe I miscalculated something earlier. Wait, so in the first method, I had n=14, so 14/2=7, 7*(105+196)=7*201=1407. Wait, but 105+196 is 201? Wait, 105+196=301? Wait, 100+200=300, 5+96=101, so total 401? Wait, no, wait 105+196: 100+196=296, plus 5 is 301. Oh, right, 105+196=301. So 7*301=2107. So that's correct. So in my initial calculation, I must have miscalculated 105+196 as 201, which is wrong. So that's where the mistake was. So the correct sum is 2107, not 1407. Wait, so that means my first method was wrong because of an arithmetic error, but the second method gives 2107.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now that I have n, a1, and an, I can plug these into the sum formula. So Sum = (14/2)*(105 + 196). Let's compute that.

<prompt_3.1> Calculate the sum of the arithmetic sequence using the formula: Sum = (n/2)*(a1 + an). </prompt_3.1>
-----

But let me cross-validate this result with another approach to be absolutely sure.

Alternative approach: list all the terms and add them up. But since there are 14 terms, that might take time, but let's try.

<prompt_3.2> Cross-validate the sum by listing all terms and adding them up. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
14 divided by 2 is 7. Then 105 + 196 is 301. So 7*301. Let's compute 7*300=2100, plus 7*1=7, so total is 2100 + 7 = 2107. So the sum should be 2107.
</execution_3.1>

<execution_3.2>
The terms are:

15*7=105

16*7=112

17*7=119

18*7=126

19*7=133

20*7=140

21*7=147

22*7=154

23*7=161

24*7=168

25*7=175

26*7=182

27*7=189

28*7=196

Now, let's add them step by step:

Start with 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, so adding each term individually gives a total of 2107, which matches the result from the formula. So that's correct. Therefore, the initial mistake was in the arithmetic when I thought 105 + 196 was 201, which is incorrect. It's actually 301. So the correct sum is 2107.

Therefore, the answer should be 2107.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, let me verify this. Let's check with the second term.

<prompt_4.1> Verify the sum by checking individual terms and their contributions. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Each term increases by 7, so the average term is (105 + 196)/2 = 201/2 = 100.5. Wait, 105 + 196 is 200 + 5 + 96 = 200 + 101 = 301. Wait, no, 105 + 196: 100 + 196 = 296, plus 5 is 301. So average term is 301/2 = 150.5. Then, multiplying by the number of terms, 14: 150.5*14. Let's compute that again. 150*14 = 2100, 0.5*14=7, so total is 2100 +7=2107. So that's correct.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
But just to be thorough, let me check with another approach. Let's use the formula for the sum of an arithmetic series again, but this time verify each step.

<prompt_5.1> Verify the sum using the arithmetic series formula again and check for errors. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
First term (a1) = 105, common difference (d) =7, number of terms (n)=14. The last term (an) = a1 + (n-1)*d = 105 + 13*7 = 105 +91=196. Correct. Then sum S = n*(a1 + an)/2 =14*(105 +196)/2=14*(301)/2=7*301=2107. Correct.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the sum of all the multiples of 15 between 100 and 200 is $\boxed{2107}$.
</Final_answer>
</think>