Consider the sequence defined recursively by $x_1$=$\sqrt2$ and where $x_n$=$\sqrt2$ + $x_n$$_-$$_1$.
Find a explicit formula for the $n^t$$^h$ term.
I considered using the general equation to find an explicit formula for any term in an arithmetic sequence. a$_n$ = a$_1$ + $d(n-1)$, but I came to no conclusion helping my argument.
Am I using the correct method?
Best Answer
Here is an approach.
$$ x_{n+1}-x_{n}=\sqrt{2} \implies \sum_{i=0}^{n-1}( x_{i+1}-x_{i}) = \sqrt{2}\sum_{i=0}^{n-1}1 $$
$$ \implies x_n-x_0=\sqrt{2} n .$$