Forgive me for any mistake in the proposal of the problem
Problem
If $\sqrt{t}+\sqrt{t+1}+\sqrt{t+2}+\sqrt{t+3}+\cdots +t=x$. Find the value of $t$ in terms of $x$
I have tried doing it with some examples but couldn't get a simple form I have also tried by squaring every number and then factoring it to get an equation in $t$ and then taking the result as $Y$ and then trying to establish a relation between $x,Y$ but I couldn't do so.
Please help me.Any help is welcome. Thanks in advance
Let be $$ n(x):=\sup_{i\in\mathbb{N}} \Big\{\sum_{k\leq i}\sqrt{t+k} \leq x\Big\} $$ (there is an obvious finite algorithm to compute it, given $t$ and $x$), then the solution will be obtained by solving $$ t^2-t = n(x)$$ Ps: $n(x)$ need $t$ and $x$ to be computed, but the question was expressing $t$ in terms of $x$, and I think this is an answer (maybe there is a better one, though).