KnowledgeBoat Logo
|

Mathematics

Simplify :

3×9n+1−9×32n3×32n+3−9n+1\dfrac{3 \times 9^{n + 1} - 9 \times 3^{2n}}{3 \times 3^{2n + 3} - 9^{n + 1}}

Indices

10 Likes

Answer

Simplify the expression :

⇒3×9n+1−9×32n3×32n+3−9n+1=3×(32)n+1−9×32n3×32n.33−(32)n+1=3×32(n+1)−9×32n81.32n−32(n+1)=3×32n+2−9×32n81.32n−32n+2=3×32n×32−9×32n81.32n−32n.32=32n(3×32−9)32n(81−32)=27−981−9=1872=14.\Rightarrow \dfrac{3 \times 9^{n + 1} - 9 \times 3^{2n}}{3 \times 3^{2n + 3} - 9^{n + 1}} = \dfrac{3 \times (3^2)^{n + 1} - 9 \times 3^{2n}}{3 \times 3^{2n}.3^3 - (3^2)^{n + 1}} \\[1em] = \dfrac{3 \times 3^{2(n + 1)} - 9 \times 3^{2n}}{81.3^{2n} - 3^{2(n + 1)}} \\[1em] = \dfrac{3 \times 3^{2n + 2} - 9 \times 3^{2n}}{81.3^{2n} - 3^{2n + 2}} \\[1em] = \dfrac{3 \times 3^{2n} \times 3^2 - 9 \times 3^{2n}}{81.3^{2n} - 3^{2n}.3^2} \\[1em] = \dfrac{3^{2n}(3 \times 3^2 - 9)}{3^{2n}(81 - 3^2)} \\[1em] = \dfrac{27 - 9}{81 - 9} \\[1em] = \dfrac{18}{72} \\[1em] = \dfrac{1}{4}.

Hence, 3×9n+1−3×32n3×32n+3−9n+1=14\dfrac{3 \times 9^{n + 1} - 3 \times 3^{2n}}{3 \times 3^{2n + 3} - 9^{n + 1}} = \dfrac{1}{4}.

Answered By

7 Likes

Related Questions