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Mathematics

Simplify :

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

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Answer

Given,

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

Simplifying the expression :

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

Hence, 3×(27)n+1+9×3(3n−1)8×33n−5×(27)n=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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