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Mathematics

Simplify :

3×27n+1+9×33n−18×33n−5×27n\dfrac{3 \times 27^{n + 1} + 9 \times 3^{3n - 1}}{8 \times 3^{3n} - 5 \times 27^n}

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Answer

Simplifying the expression

⇒3×27n+1+9×33n−18×33n−5×27n=3×(33)n+1+(32)×33n−18×33n−5×(33)n=3×33(n+1)+32+3n−18×33n−5×33n=31+3(n+1)+33n+133n(8−5)=31+3n+3+33n+13×33n=33n+1.33+33n+133n+1=33n+1(33+1)33n+1=33+1=27+1=28.\Rightarrow \dfrac{3 \times 27^{n + 1} + 9 \times 3^{3n - 1}}{8 \times 3^{3n} - 5 \times 27^n} = \dfrac{3 \times (3^3)^{n + 1} + (3^2) \times 3^{3n - 1}}{8 \times 3^{3n} - 5 \times (3^3)^n}\\[1em] = \dfrac{3 \times 3^{3(n + 1)} + 3^{2 + 3n - 1}}{8 \times 3^{3n} - 5 \times 3^{3n}} \\[1em] = \dfrac{3^{1 + 3(n + 1)} + 3^{3n + 1}}{3^{3n}(8 - 5)} \\[1em] = \dfrac{3^{1 + 3n + 3} + 3^{3n + 1}}{3 \times 3^{3n}} \\[1em] = \dfrac{3^{3n + 1}.3^3 + 3^{3n + 1}}{3^{3n + 1}} \\[1em] = \dfrac{3^{3n + 1}(3^3 + 1)}{3^{3n + 1}} \\[1em] = 3^3 + 1 \\[1em] = 27 + 1 \\[1em] = 28.

Hence, 3×27n+1+9×33n−18×33n−5×27n=28\dfrac{3 \times 27^{n + 1} + 9 \times 3^{3n - 1}}{8 \times 3^{3n} - 5 \times 27^n} = 28.

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