1 + 2 + 4 + 8 + ⋯
Hey there kiddo! Have you ever played with blocks and stacked them on top of each other? Well, imagine if we have blocks that keep doubling its size every time we add another one. For example, our first block is size 1, our second block is size 2, our third block is size 4, and so on.
Now, let’s say we want to know how many blocks we have stacked up in total. How do we figure it out? We could count each individual block one by one, but that could take a long time. Instead, we can use a special math trick!
This trick is called “infinite series,” which means that we keep adding blocks forever and ever, without ever stopping. But don’t worry, we will only add up a certain number of blocks for now.
We start by adding our first block of size 1 to our second block of size 2, which gives us a total of 3. Then, we add our third block of size 4 to our total, which gives us 7. We keep going and add our fourth block of size 8 to our total, which gives us 15. Do you see how we are getting bigger and bigger numbers as we add more blocks?
Now, here comes the tricky part. We want to figure out how many blocks we would have if we added them up forever and ever, without ever stopping. Sounds impossible, right? Well, it turns out that there is a special formula we can use to figure it out!
The formula is:
Total = (first term)/(1 – common ratio)
In our case, our first term is 1, and our common ratio is 2 (which means we keep doubling the size of each block). So, we have:
Total = 1/(1 – 2)
Total = 1/-1
Total = -1
Wait a minute, negative one? That can’t be right! But surprisingly, it is. This means that if we were to add up infinitely many blocks that keep doubling in size, our total would be -1.
Now, why is this? It has to do with something called “convergence,” which means that even though we are adding up an infinite amount of numbers, they eventually start to get smaller and smaller until they approach a certain value. In our case, that value just happens to be -1.
So, there you have it kiddo! The math behind adding up blocks that keep doubling in size. Pretty cool, huh?
Now, let’s say we want to know how many blocks we have stacked up in total. How do we figure it out? We could count each individual block one by one, but that could take a long time. Instead, we can use a special math trick!
This trick is called “infinite series,” which means that we keep adding blocks forever and ever, without ever stopping. But don’t worry, we will only add up a certain number of blocks for now.
We start by adding our first block of size 1 to our second block of size 2, which gives us a total of 3. Then, we add our third block of size 4 to our total, which gives us 7. We keep going and add our fourth block of size 8 to our total, which gives us 15. Do you see how we are getting bigger and bigger numbers as we add more blocks?
Now, here comes the tricky part. We want to figure out how many blocks we would have if we added them up forever and ever, without ever stopping. Sounds impossible, right? Well, it turns out that there is a special formula we can use to figure it out!
The formula is:
Total = (first term)/(1 – common ratio)
In our case, our first term is 1, and our common ratio is 2 (which means we keep doubling the size of each block). So, we have:
Total = 1/(1 – 2)
Total = 1/-1
Total = -1
Wait a minute, negative one? That can’t be right! But surprisingly, it is. This means that if we were to add up infinitely many blocks that keep doubling in size, our total would be -1.
Now, why is this? It has to do with something called “convergence,” which means that even though we are adding up an infinite amount of numbers, they eventually start to get smaller and smaller until they approach a certain value. In our case, that value just happens to be -1.
So, there you have it kiddo! The math behind adding up blocks that keep doubling in size. Pretty cool, huh?
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