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we're told consider the sum two plus five plus eight plus 11 which expression is equal to the sum above and they tell us choose all answers that apply so like always pause the video and see if you can work through this on your own so when you look at the sum it's clear you're starting it two you're adding three each time and we also are dealing with you have four total terms now we could try to construct an equation here or an expression using Sigma notation but instead what I like to do is look at our options we really only have to look at these two options here and expand them out what sum would be each of these B well this is going to be the same thing as we're starting at N equals one so this is going to be three times when N equals 1 3 times 1 minus 1 and then plus then it'll go to 2 N equals 2 3 times 2 minus 1 then we're going to go to N equals 3 so plus 3 times 3 minus 1 and then finally we're going to go to N equals 4 so plus 3 times when N equals 4 this is 3 times 4 minus 1 so just to be clear this is what we did when N equals 1 let me write it down N equals 1 this is what we got when N equals 2 this is what we got when N equals 3 and this is what we got when N is equal to 4 and we stopped at N equals 4 because it tells us right over there we go we start at N equals wanted to go all the way to N equals 4 so what is this equal so let's see 3 times 1 minus 1 is 2 so this is looking good so far 3 times 2 minus 1 that's 6 minus 1 that is 5 so still looking good 3 times 3 minus 1 that's 9 minus 1 once again still looking good 3 times 4 minus 1 that is 11 so we like this choice so I would definitely select this one now let's do the same thing over here when N is equal to Z it's going to be 2 plus 3 times 0 so that's just going to be 2 and then plus when n is equal to 1 it's going to be 2 plus 3 times 1 which is 5 this is starting to look good now when n is equal to this was N equals 0 this was N equals 1 so now we're at N equals 2 so at N equals 2 2 plus 3 times 2 is 2 plus 6 which is 8 and this makes sense every time we increase n by 1 we are adding another 3 which is consistent what we saw there we start at 2 when N equals 0 this is just that 3 n is just 0 so you start it two and then you keep every time you increase n by 1 you are just adding 3 again so finally when n is equal to 3 2 plus 3 times 3 is 2 plus or is 11 I should say and so this also is exactly the same sum so I feel good about both of these let's do one more example so we're here we were given the sum and we're saying choose one answer which of these is equivalent to this sum right over here well like we did before let's just expand it out and what's different here is that we just have a variable but that shouldn't make it to much more difficult so let's do the situation and I'll write it out let's do the situation when n is equal to 1 when n is equal to 1 it's going to be K over 1 plus 1 K over 1 plus 1 I'll write it out this is n is equal to 1 and then plus when n is equal to 2 is going to be K over 2 plus 1 this is n is equal to 2 and then we're going to keep going when n is equal to 3 it's going to be K over 3 plus 1 that's N is equal to 3 and then plus finally because we stopped right over here at N equals 4 when N equals 4 it's going to be K over 4 plus 1 that's when n is equal to 4 so this is all going to be equal I'll just write it over here this is equal to K over 2 k over 3 plus K over 4 plus K over 5 which is exactly this choice right over right over here and actually if I had looked at the choices ahead of time I might have even been able to save even more time by just saying well look actually if you just try to compute the first term when n is equal to 1 it would be K over 2 well only this one is starting with the K over 2 this this one has no case here which is sketchy they're trying to look at the error where you try to replace the K with the number as well not just the end so that's what they're trying to do here here they're trying to let's see well this isn't as obvious what they're even trying to do what right over here we put the K in the denominator and here if you swapped the N in the case then you would have gotten this thing right over here so we definitely feel good about choice a

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