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nth Term of a Sequence Find a formula for the $n$ th term of the sequence

$$

\sqrt{2}, \quad \sqrt{2 \sqrt{2}}, \quad \sqrt{2 \sqrt{2 \sqrt{2}}}, \quad \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2}}}}, \ldots

$$

[Hint: Write each term as a power of $2 . ]$

$a_{n}=2^{\frac{2^{n}-1}{2^{n}}}$

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So we have a sequence kind of interesting looking. And look what happens when you go from this one to this one. It's like they multiply that number by by two. But then they square with the answer and then they take this number and they multiply it by two mhm and then they square root it and then the next one they take that answer mhm and they multiply it by two and then they square root it. So we have to find a pattern. So let's look, I'm going to start this going and I'm gonna do it this way. We know that if we're having about the which term this is so the first term we know is two to the one half power. The second term is let's figure out what power of two this is. So this is inside here. This is two times two to the one half, and all of that is to the one half power. So this is 2/2. So inside here we have to to the three halves power to the one half. And when you take forward, two of our you multiply, so this will become two to the 3/4 power. Now the third one, the third one. We know that this part right here is that two to the three force power and then it's being multiplied by two. And then it's being square rooted. So we know that this is, um, this is to the 4/4 power that's to the first power. And so inside. We have to to the seven forest power, because when we multiply like basis, we add the exponents. When we add fractions, we only add the numerator as we don't add the denominators. And then that's to the one half power. So when we take the power to a power, we multiply. And this becomes, too to the 7/8 and let's do one more. Well, I guess we only have one more to do. We could could write more. So now we know this part right here. We'll move that down. This part right here is that to to the 7/8 and then it's being multiplied by two. So I'm going to write it as 8/8 and then it's being square rooted. So to the one half power. So inside here we have to to the eight plus seven is 15 8th, and then that's to the one half power. So this becomes take a power to a power. This becomes too to the 15/16 and notice that the exponent the numerator is always one power less than the denominator. But also noticed, this is two to the first power. This is two to the second power. This is two to the third power. This is to the fourth power. So we notice a pattern and we can write our and return so two to the power of and the denominator is to to the end. And the numerator is one less than that. So two to the N minus one. So there's the entire exponents. That would be, um, for each of these powers. So there is our end through all mm

Michigan State University

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