Let me give you a totally different sequence of numbers as an example of a sequence which gets closer and closer to a certain number. How about the sequence of \(p_n\) such that

$$\begin{aligned} p_1 &= 3 \\ p_2 &= 3.1 \\ p_2 &= 3.14 \\ p_3 &= 3.141 \\ p_4 &=3.1415 \ldots \\ \end{aligned} $$As \(n\) gets bigger, \(p_n\) gets closer to \(\pi!\)

Similarly, in the question to Day 2: Challenge (4 of 4), the \(a_n\) get closer and closer to \(\sqrt{2} \approx 1.414 \ldots. \)

All of the terms (we call them \(a_n\) because \(n\) can be any whole number) get closer and closer to the square root of \(2\), so \(a_2\) is closer to \(\sqrt{2}\) compared to \(a_1,\) and \(a_3\) is closer to \(\sqrt{2}\) compared to \(a_2.\) Does that seem weird?

You can see this pattern by calculating the decimal value of the fractions:

$$\begin{aligned} a_1 &= \frac{3}{2} = 1.5 \\ a_2 &= \frac{17}{12} = 1.41\overline{666}\\ a_3 &= \frac{577}{408} = 1.41421568627 \\ \end{aligned} $$Do you see that as the \(n\) in the \(a_n\) gets bigger and bigger, the decimal gets closer and closer to \(1.41421356237 \ldots,\) which equals \(\sqrt{2}?\)

Now, the question asks you to calculate these fractions yourself, by plugging in a previous value of \(a_n\) into

$$ a_{n+1} = \frac{a_n + \frac{2}{a_n}}{2}.$$

Start with \(a_0,\) which we are given is equal to \(1.\) (Yes, it's a bit weird to start with zero, but this is a very common thing to see in mathematics!) Putting \(a_0 = 1\) into the formula, we can find out \(a_1:\)

$$ a_1 = \frac{a_0 + \frac{2}{a_0}}{2} \implies a_1 = \frac{1 + \frac{2}{1}}{2} = \frac{3}{2} = 1.5 $$

Now, to get the next number in this sequence, which also brings us closer to \(\sqrt{2},\) plug this value of \(a_1\) into the formula again, to get \(a_2,\) and you can plug in the value of \(a_2\) into the formula to get the next value after that, \(a_3.\) You can keep going and going, until you get to \(a_{100},\) and then you can even keep going, after that, with your number still becoming more and more close to \(\sqrt{2}!\)

Is it a coincidence that this formula happens to give a series of numbers that gets closer and closer to \(\sqrt{2}?\) Can we prove why this formula works? Where did this formula come from, anyway? These questions, and this neat fact, is what makes mathematics really fascinating! 🙂

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