problem: If $x+\frac{1}{x}=7$, then what is the value of $x^{2}+\frac{1}{x^{2}} + 1$?
solution: Squaring the equation provided, we get $x^2+2(x)\left(\frac{1}{x}\right) +\frac{1}{x^2} = x^2 + 2 + \frac{1}{x^2}=49,$ so $x^2+\frac{1}{x^2} + 1=\boxed{48}.$