Sums of consecutive integers

the sum of four consecutive odd integers is 136 what are the four integers so before even attempting to tackle it let's let's think about what it means to be a consecutive odd integer what four consecutive odd integers could be referring to so we could start with let's say we had three the next odd integer so we want consecutive honors before what the next odd integer would be five then the one after that would be seven and then one after that would be nine these are consecutive odd integers another example we could start at 11 then the next odd integer is 13 the next one is 15 the next one is 17 the example of non consecutive odd integers if someone went from three straight to seven these are not consecutive the next integer the next odd integer I should say after three is five not seven so these are examples of consecutive odd integers these are not consecutive odd integers so with that out of the way let's actually try to tackle this question and I encourage you to pause this video right now and try to tackle this on your own before I take a go at it well as you can imagine a little bit of algebra might be handy here so let's let's let X define let's let X be equal to the smallest smallest of the four of the four so if X is the smallest of the four integers of the four consecutive odd integers if X is a smallest out of them how can we express the other three in terms of X well just use this existing examples right over here if that is X then how would we express this in terms of X well this would be X plus two the next the next odd integer is going to be the previous odd integer plus two you're essentially skipping the next number which is going to be even so if you added one you just get to an even number so you have to add two to get to the next odd one well if you add two again now if you add two to X plus 2 well now you get two x plus four you add two to that you get two x plus six and you see that again right over here if this was X then this is X - this is X plus 4 and this is X plus 6 so in general if X is the smallest of the integers we can define the other 3 as X plus 2 X plus 4 and X plus 6 and so we let's take the sum of them and set that equal to 136 and then solve for X so the smallest one is X now the next smallest one is going to be X plus 2 now the next the one after that's going to be X plus 4 and then the one after that is going to be X plus 6 so this is the sum of the four consecutive odd integers and they tell us that that is going to be equal to 136 this is equal to 136 and now we can just literally solve for X we have just one unknown here so let's let's add these X terms together so we have 1 X 2 X is 3 X is 4 X's so we can rewrite those literally as 4 X and then we have 2 plus 4 which is 6 plus another 6 is 12 4 X plus 12 is equal to 136 so to solve for X a good starting point would be to just to isolate the X terms on one side of the equation or try to get rid of this 12 well to get rid of that 12 we'd want to subtract 12 from the left-hand side but we can't just do it from the left-hand side then this equality wouldn't hold anymore if these things do things were equal before subtracting the 12 well then we got to if we want to keep them equal if we want the left and the right to stay equal we got to subtract 12 from the from both sides so subtracting 12 from both sides gives us well on the left-hand side we're just left with 4 X and on the right hand side we are left with 136 minus 12 is 100 and 120 120 4 I do that right yet 124 so what is X well we just divide both sides by 4 you can divide both sides by 4 by 4 to solve for X and we get X as being equal to then the same original color X is equal to 124 divided by 4 so let's see 100 divided by 4 is 20 five 24 divided by 4 is 6 25 plus 6 is 31 if you don't feel like doing that in your head you could also of course do traditional long division goes into 120 4 4 goes and doesn't go into 1 4 goes into 12 3 times 3 times 4 is 12 you subtract bring down the next 4 4 goes into 4 one time you get no remainder so X is equal to 31 so X is the smallest of the four integers so this right over here X is 31 X plus 2 is going to be 33 X plus 4 is going to be 35 and X plus 6 X plus 6 is going to be 37 so our four integers are four consecutive odd integers are 31 33 35 and 37