Find 1+2+3+4+...+ 100, the sum of the first 100 natural number...

Question

Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.

Accepted Answer

Hello. Today we are going to find the total sum of the 49 multiples of five. So we are given the sequence of five plus 10 plus 15 plus all the numbers all the way to 245. Now, before we begin this, let's go ahead and check to see if this is an arithmetic sequence or not. The first term is five. And in order to go to the next term 10, we're going to add 5-5. And in order to go from 10 to 15 we're going to add 5-10. So here we have a common difference of five in each of the terms. So this confirms that this is an arithmetic sequence with the common difference of five. Since we are dealing with an arithmetic sequence, we have an equation that allows us to find the sum of an arithmetic sequence. That equation is the sum s is equal to n over two times the quantity A sub one plus a sub N. Here a sub one is equal to your first term in the arithmetic sequence and a sub n is equal to the last term of the arithmetic sequence. So let's go ahead and start by labeling a sub one a sub N. Since a sub one is the first term, the first term is going to be five. So we know that a sub one is equal to five. Next a sub N is the last term of the arithmetic sequence. Since we are told that there are 49 multiples in the sequence. We can go ahead and assume that 245 is going to be the last term in the sequence. So a sub N is going to equal to 245. Now, what about n? Since we end represents the total amount of terms that are within the arithmetic sequence. And since we are asked to find the sum of the 1st 49 multiples, we're going to allow N To equal to 45. So let's go ahead and plug in these values to the some of the arithmetic sequence. The sum is going to be s is equal to end which is 45/2 times the quantity a sub one, which is five plus a sub N which is Five plus 245 is going to give us 250. So what we have here is 45 times 250. All divided by two. Now 45 times 250 is going to give us 12,250. And dividing that by two is going to give us 6125 and this is going to be the sum of the given arithmetic sequence. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to find the sum of an arithmetic sequence and I'll go ahead and see you all in the next video.

Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.

Key Concepts

Arithmetic Series

Formula for the Sum of the First n Natural Numbers

Natural Numbers

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