<p>The sum of the positive odd integers less than 100 is subtracted from the sum of the positive integers less than 100. What is the resulting difference?</p>
<p>Can I apply this from Godot:</p>
<p>sum of an arithmetic sequence = (first term + last term)/2 * (number of terms)</p>
<p>You're taking away all of the odd numbers from the set of all numbers from 1 to 100. This will leave you with anything that isn't odd--that is, all even integers from 1 to 100. So your numbers will be 2, 4, 6, 8, ..., 100. At this point, you can apply the handy formula for the sum of an arithmetic sequence, if you know it--the sum is the average of the first and last terms multiplied by the number of terms. You initially had 100 integers, and removed the odd ones (half of the numbers were odd), so you now have 50 integers, with the first being 2 and the last being 100. The sum, then, is 50<em>(2+100)/2 = 50</em>51 = 2550.</p>
<p>I understand, thanks if I strip away all odd number, I have left 2,4,6,8,10...100
The first term in this sequence is 2 and the last term is 100
Applying: 50*(2+100)/2 = 2550.</p>
<p>How about the odd term:
I strip away all even number, what I have left is 3,5,7,9,11...99 right, the first term in this sequence is 3 and the last one is 99. Using the concept above I have ...</p>
<p>Wait, ScythianEmpire, can you explain how do I figure out the number of term?</p>
<p>If you strip away all the even numbers, you'll also have 1 left as well, so your numbers will start with 1 and end with 99.</p>
<p>There's another little formula that can help you find the number of terms in an arithmetic sequence, if you are having difficulty figuring it out through other means. It goes like this: the number of terms is equal to one greater than the difference of the last term and the first term divided by the common difference. Applied to the initial example (with even numbers only), this will give you (100-2)/2 + 1 = 50. Applied to your second question about the odd numbers only, you'll have (99-1)/2 + 1 = 50. If you're just looking at the whole set of positive integers from 1 to 100, it would be (100-1)/1 + 1 = 100.</p>
<p>(For clarification: the "common difference" in an arithmetic sequence is the difference between any two numbers that are next to each other in the sequence...so in the example about the odd numbers 1, 3, 5, 7, ..., 99, the common difference will be 3-1 = 2, or 5-3 = 2, and so on)</p>
<p>Thanks a lot, so the answer to the question above is...50, right??
Can you explain a little bit more, I appreciate a lot Scythian...(Wait, are your name referring to one of the most fearsome archers during Ancient time, or perhaps I get mixed up with the Hitties :) I love History though)</p>
<p>I have a question, why (99-1)/2 + 1 but not (99-2)/2+1, what does this 1 stand for, I did not get when I review the concept, thanks</p>
<p>Wait, which 1 are you referring to? The 1 being subtracted, or the 1 being added?</p>
<p>If you mean the former, the number that you're subtracting is the first term of the sequence. In the example, the sequence begins with 1 and ends with 99, so the subtraction involves (99-1).</p>
<p>Oh, I get you</p>
<p>since the even series started with 2 and end with 100, so 100 - 2
for the odd series, it started with 1 and end with 99, so 99 - 1, is that correct?</p>
<p>Yeah, that's right.</p>
<p>OK, for example if there is a series as following
n, n + 3, n + 5, n + 7... n + 99</p>
<p>What I have to find out about the number of term is to take n + 99 - n/ 2 + 1, is that correct in this case??</p>
<p>For it to be an arithmetic sequence, each term has to differ by a constant amount, so the first term in that case would have to be n + 1, not n. Then the number of terms would be ((n + 99) - (n + 1)) / 2 + 1.</p>
<p>Here, try this example: How many terms are in the sequence 5, 8, 11, 14, ..., 35?</p>
<p>35 - 5/5+1 = 5, right?</p>
<p>Not quite...remember that you are dividing by the common difference in the equation, which in this example is 3 (that is, each term is 3 greater than the term before it). So you have (35 - 5)/3 + 1 = 11.</p>
<p>One more example, maybe? How many terms are in the sequence 3, 7, 11, 15, ..., 47?</p>
<p>calm down, ok?</p>
<p>I know that you're taking the test tomorrow (so am I), but just relax for an hour. :) we'll be fine.</p>
<p>47 - 3/ 4 + 1 = ?? 3 + 4 = 7, 11 + 4 = 15?</p>
<p>Oh, I understand (47 - 3)/4 + 1 = 12. Get you!</p>