What walk 1 + 2 + 3 + … + n equal? uncover How to Quickly include the Integers indigenous 1 to n out how to admire your girlfriend at parties by conveniently calculating the amount of all the integers from 1 up to any type of number they choose.

You are watching: Sum of first 100 positive integers

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In the last episode, us learned an impressive trick that you can use come quickly include up all the integers indigenous 1 to 100. And also that yes, really was no little feat due to the fact that we rotate the herculean task of performing 100 addition problems—that is including up 1 + 2 + 3 + 4 + … + 100—into a cute fuzzy kitten the a single multiplication problem. Return this trick is undeniably impressive, it’s not exactly the kind of thing you can pull the end at next to admire your friends since they could case that you simply memorized the answer.Which might lead you come wonder: rather of just including up the very first 100 positive integers, is there a way to quickly calculate the amount of the first 50, 200, or maybe also 1,000 hopeful integers? In other words, is over there a means to quickly calculate the sum of all the integers native 1 up to any kind of other number—which we’ll call “n”—that your friends can throw at you? That would be a rather outstanding trick, right? Well, as luck would have it, over there is a means to carry out it…and that’s precisely what we’re going come talk about today.

## Recap: including the Integers native 1 to 100

Before we figure out exactly how to add up every the integers native 1 to n, let’s recap exactly how to add up all the integers from 1 come 100. The crucial to this is our girlfriend the associative residential or commercial property of enhancement which claims that you are cost-free to add together a team of number in any type of order you like. In the past, we’ve seen just how this liberty can be provided to assist you execute lightning quick mental addition, and now this exact same property comes to the rescue again since it method that we’re totally free to include up every the number from 1 come 100 in pairs.In particular, we desire to type pairs comprise one number indigenous the start of the sequence and one number native the end: 1+100, 2+99, 3+98, and also so on. Why does that help? because each of these pairs of numbers adds as much as 101. And since there are 50 together pairs, us can really quickly number out—without doing every 100 addition problems—that the amount of the first 100 hopeful integers is 50 x 101 = 5,050.

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