What time is 5 1/2 after 9:22 pm
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I suppose denotes the n-th term of some sequence, and we're given the 3rd and 5th terms
and
. On this information alone, it's impossible to determine the 100th term
because there are infinitely many sequences where 2 and 16 are the 3rd and 5th terms.
To get around that, I'll offer two plausible solutions based on different assumptions. So bear in mind that this is not a complete answer, and indeed may not even be applicable.
• Assumption 1: the sequence is arithmetic (a.k.a. linear)
In this case, consecutive terms <u>d</u>iffer by a constant d, or
By this relation,
and by substitution,
We can continue in this fashion to get
and so on, down to writing the n-th term in terms of the first as
Now, with the given known values, we have
Eliminate to solve for d :
Find the first term :
Then the 100th term in the sequence is
• Assumption 2: the sequence is geometric
In this case, the <u>r</u>atio of consecutive terms is a constant r such that
We can solve for in terms of
like we did in the arithmetic case.
and so on down to
Now,
Eliminate and solve for r by dividing
Solve for :
Then the 100th term is
The arithmetic case seems more likely since the final answer is a simple integer, but that's just my opinion...