Given the sequence 1, 2, 4, 7, 11... and the rule: each term is the previous term plus the previous term's term number. For instance, the 51st term of this sequence is equal to the 50th term plus 50. Find the 100th term of this sequence.
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When you take a look at the sequence 1, 2, 4, 7, 11..., you'll notice that the difference between the first and second term is 1, the second and third is 2, the third and fourth is 3, and etc. In order to construct the second term, you add 1, and to construct the third, you add 2. Our goal is to construct the 100th term, and that would be the sum of 1 and the natural numbers from 1 to 99 (inclusive). Finding the sum of the natural numbers between 1 and 99 (inclusive) is simply the sum of the first and last terms (1+99=100), multiplied by half of the number of natural numbers within 1 to 99. Finally, add the 1 like I mentioned earlier to the sum of the first 99 natural numbers, and we get 4 9 5 1 .