\(_{u_k=u_{k-1}+4\left(k-1\right)+3=u_{k-2}+4\left(k-2\right)+4\left(k-1\right)+2.3=.....=u_1+4\left(1+2+3+...+k-1\right)+3\left(k-1\right)=\left(k-1\right)\left(2k+3\right)}\)
\(\Rightarrow lim\dfrac{\sqrt{u_{kn}}}{n}=\dfrac{\sqrt{\left(2kn+3\right)\left(kn-1\right)}}{n}=\sqrt{\left(2k+\dfrac{3}{n}\right)\left(k-\dfrac{1}{n}\right)}=k\sqrt{2}\)
\(\Rightarrow lim_P=\dfrac{1\sqrt{2}+4\sqrt{2}+4^2\sqrt{2}+...+4^{2022}\sqrt{2}}{1\sqrt{2}+2\sqrt{2}+2^2\sqrt{2}+....+2^{2022}\sqrt{2}}=\dfrac{4^{2023}-1}{3.\left(2^{2023}-1\right)}=\dfrac{2^{2023}+1}{3}\)
\(\Rightarrow a+b-c=0\)
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