@placidmacaw-0 Hello! The variables in this question might look a little bit strange, because they have subscripts, like \( a_1, a_2, a_3,\) etc. We're used to having variables without subscripts, so you might wonder, "Why do we need the little numbers on the bottom?" The little numbers are really useful, because they signify that there is a pattern to all of the \(a's.\)

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!