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連續單數的和 (英) : 連續單數的和
- We're told that the sum of three consecutive
- odd integers is 231.
- What is the largest integer?
- So let's think about this a little bit.
- Let's say that x is the smallest integer.
- x is equal to the smallest odd of these three.
- It's not the smallest odd integer of all integers, it's
- is the smallest odd of these three, the
- smallest odd integer.
- So what's the next one going to be?
- Well, if I have one odd integer, what's going to be
- the next odd integer?
- Let's think about this.
- If x was 3, what's the next odd integer?
- It's 5.
- And then what's the next one after that?
- It's 7.
- And the next one after that?
- It's 9.
- So every time we add 2.
- So if the smallest one is x, the next smallest odd integer
- is x plus 2, is equal to the next smallest odd integer--
- I'll write integer here-- odd integer.
- And then what would be the next one?
- Well, we're going to add 2 to this one, right?
- So it's going to be x plus 4.
- Think about it.
- If the smallest is 3, then you have x plus 2, which is 5.
- And then you have x plus 4, which is 7.
- So this will be the largest of the consecutive odd integer in
- this group.
- And they tell us that the sum of these consecutive odd
- integers is 231.
- What is the largest integer?
- So if I take x, x plus 2 and x plus 4, and I sum them, they
- should be equal to 231.
- So let's do that.
- So we have x plus x plus 2, plus x plus 4, and this needs
- to be equal to 231.
- And when they ask us what's the largest one, we're going
- to have to tell them what x plus 4 is equal to.
- So let's just solve this equation.
- So let's add our x terms. We have one x, two x, three x's,
- so we get 3x plus-- and then what are our constants?
- We have a 2 and we have a 4.
- So 3x plus 6 is equal to 231.
- Now, let's get rid of the 6 from the
- left-hand side of the equation.
- The best way to do that it is to subtract 6 from both sides.
- So let's subtract 6 from both sides.
- The left-hand side, we're just left with the 3x.
- The 6's cancel out.
- The right-hand side, 231 minus 6 is 225.
- We have 3x is equal to 225.
- To isolate the x, let's just divide both sides by 3.
- The left-hand side, the 3's cancel out-- that was the
- whole point behind dividing by 3-- we get just an x is being
- equal to-- and 225 divided by 3.
- Let me do it over here.
- So 3 goes into 225.
- It goes into 22 7 times.
- 7 times 3 is 21.
- 22 minus 21 is 1.
- Bring down the 5.
- 3 goes into 15 5 times.
- 5 times 3 is 15.
- Subtract, no remainder.
- So 225 divided by 3 is 75.
- So the smallest, the smallest of the odd integers is 75.
- So this one is 75.
- What's x plus 2 going to be equal to?
- Well, that's going to be 2 more, 77.
- And what's x plus 4 going to be equal to?
- Well, that's the largest of them.
- x is 75 plus 4 is going to be 79.
- And notice, we have three odd integers.
- They're consecutive.
- They're the, you know, they're the odd integers that come
- directly after each other.
- And let's verify that when we add them up, we get 231.
- So if we get 75 plus-- let me just write it like this-- 75
- plus 77, plus 79, want to add them all up.
- 5 plus 7 is 12.
- 12 plus 9 is 21.
- Carry the 2.
- 2 plus 7 is 9.
- 9 plus 7 is 16.
- 16 plus 7 is 23.
- So there you have it.
- The three consecutive odd integers, when you add them
- up, you got 231.
- They're consecutive and odd.
- They ask, what's the largest?
- The largest is 79.