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[tex] \it a = \dfrac{2^{4n}\cdot5^{2n+2} +800^n}{25^{n+1}+50^n} = \dfrac{2^{4n}\cdot5^{2(n+1)}+ (16\cdot50)^n}{25^{n+1}+50^n} = \\ \\ \\ = \dfrac{2^{4n}\cdot25^{n+1}+ (2^4\cdot50)^n}{25^{n+1}+50^n} = \dfrac{2^{4n}\cdot25^{n+1}+ 2^{4n}\cdot50^n}{25^{n+1}+50^n} = \\ \\ \\ = \dfrac{2^{4n}(25^{n+1}+ 50^n)}{25^{n+1}+50^n} = 2^{4n} =(2^4)^n = 16^n=(4^2)^n = (4^n)^2 [/tex]
