Finding nth term of a sequence (explicit formula)

In summary, the sequence is given by √2, √(2√2), √(2√(2√2)), √(2√(2√(2√2))), etc. The general rule is √2^(2n-1/2) but this rule does not work for the first term. Instead, the terms can be written in terms of fractional exponents as √2= 2^(1/2), √(2√2)= 2^(3/4), etc. The general term can be guessed as √2^(2n-1/2) and proven using induction and the fact that a_(
  • #1
demonelite123
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√2, √(2√2), √(2√(2√2) ), √(2√(2√(2√2) ) )

this sequence has been giving me a lot of trouble. i have no idea how to write the fomula. please help me.
 
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  • #2
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  • #3
well i did this so far:
√2, √2^(3/2), √2^(5/2), √2^(7/2)

and i got the general rule:
√2^(2n-1 / 2)

the only problem is that it works for everything but the first term. can someone please help me fix up my rule so that it fits all of the terms?
 
  • #4
No that doesn't work for any term, but writing them in terms of fractional exponents is the right way to go. You have [itex]\sqrt{2}= 2^{1/2}[/itex], [itex]\sqrt{2\sqrt}{2}}= (2(2^{1/2})^{1/2}= (2^{3/2})^{1/2}= 2^{3/4}[/itex], etc., [itex]\sqrt{2\sqrt{2\sqrt}{2}}}= (2(2^{3/4})^{1/2}=(2^{7/4})^{1/2}= 2^{7/8}[/itex], etc..

You should be able to guess the general term from that. And, in fact, you should be able to prove it using induction and the fact that [itex]a_{n+ 1}= \sqrt{2a_n}
 

1. What is the purpose of finding the nth term of a sequence?

The purpose of finding the nth term of a sequence is to be able to predict the value of any term in the sequence without having to list out all the previous terms. It allows for a more efficient way of solving mathematical problems and understanding patterns in a sequence.

2. How do you find the nth term of a sequence?

The nth term of a sequence, also known as the explicit formula, can be found by identifying a pattern in the sequence and then using that pattern to create an equation. This equation will then give you the value of the nth term based on the term number (n).

3. What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence is a sequence where each term is found by adding a constant value to the previous term. A geometric sequence, on the other hand, is a sequence where each term is found by multiplying a constant value to the previous term. In other words, in an arithmetic sequence, the difference between consecutive terms is constant, while in a geometric sequence, the ratio between consecutive terms is constant.

4. Can the nth term of a sequence be negative?

Yes, the nth term of a sequence can be negative. The explicit formula for a sequence can include negative numbers, and the term number (n) can also be negative, representing terms that come before the first term in the sequence.

5. Can the nth term of a sequence be a fraction or decimal?

Yes, the nth term of a sequence can be a fraction or decimal. The explicit formula for a sequence can involve operations that result in a fraction or decimal, and the term number (n) can also be a fraction or decimal, representing non-integer terms in the sequence.

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