Find the sum of the first 50 terms of the arithmetic sequence: -1...

Question

Find the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....

Accepted Answer

Hello. Today we are asked to find the sum of the 1st 30 terms of the given arithmetic sequence. Now we have an equation that allows us to find the sum of an arithmetic sequence. That is when S is equal to N over two Times a sub one plus a sub n. A sub one represents the first term in your sequence. And ace of N represents the last term of the sequence. And N represents the amount of terms that are within your sequence. Now since we are told that we want to find the sum of the 1st 30 terms we're going to allow N to equal to 30. And since a sub one represents the first number in our given sequence, the first number is negative nine. So a sub one is equal to negative nine. Now if we take a look at the given sequence, what we are given is -9 - -303. And continuing, we don't know what the last term of the sequence is. So we're going to need a sulfur it. Well we have an equation that allows us to represent the end term of the given arithmetic sequence. And that is a sub N is equal to a sub one which is your first term plus the quantity and minus one times the common difference D. So let's go ahead and figure out what the common difference is. Since we know what a sub one is looking at the terms in order to go from negative nine to negative six we're going to add a total of three to negative nine. In order to go from negative six to negative three. We're going to add three in order to go from negative 3 to 0. We're also adding three and to go from 0 to 3 we need to add three. So if you notice every term within the sequence is a difference of three. So in other words we can allow D two equal to positive three since every term is incremental by positive three. So since we have our common difference and we have our first term we can go ahead and plug this into the equation to figure out what the 30th term is. So a sub N is equal to the first term which is negative nine plus the quantity and minus one times D. Which is three. Now in order to simplify this we can go ahead and distribute three into the quantity and minus one. This is going to give us a sub N is equal to negative nine Plus three N -3. Now we just need to go ahead and combine like terms negative nine minus three is going to give us negative 12. So what we have is a sub N is equal to negative plus three N. And this equation allows us to find any term within the given arithmetic sequence. Now we are told that we want to have the 30 terms so we need to look for a sup 30. So we're going to allow N2 equal to 30 for this equation. This is going to give us a sub 30 is equal to negative 12 plus three times 30 Three times 30 is going to give us 90. So we have native 12 plus 90 And negative 12 plus 90 is going to give us 78. So the 30th term of the arithmetic sequence is going to be positive 78. Now that we have our a sub n r a sub one and n let's go ahead and plug these values into the some equation. Doing so is going to give us S is equal to end which is 30/2 times the quantity a sub one which is negative nine plus 78. Now 30 divided by two is going to give us 15 and negative nine plus 78 is going to give us 15 times 69 is going to be 1,035 and that 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 C. 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 the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....

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