Question

\(\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}\right).3^5+\left(\dfrac{1}{3^5}+\dfrac{1}{3^6}+\dfrac{1}{3^7}+\dfrac{1}{3^8}\right).3^9\)

Answer 1

\(=\dfrac{3^5}{3}+\dfrac{3^5}{3^2}+\dfrac{3^5}{3^3}+\dfrac{3^5}{3^4}+\dfrac{3^9}{3^5}+\dfrac{3^9}{3^6}+\dfrac{3^9}{3^7}+\dfrac{3^9}{3^8}\\ =2\left(3^4+3^3+3^2+3\right)\)

Answer 2

\(=\left(\dfrac{3^5}{3}+\dfrac{3^5}{3^2}+\dfrac{3^5}{3^3}+\dfrac{3^5}{3^4}\right)+\left(\dfrac{3^9}{3^5}+\dfrac{3^9}{3^6}+\dfrac{3^9}{3^5}+\dfrac{3^9}{3^8}\right)\)

\(=\left(3^4+3^3+3^2+3\right)+\left(3^4+3^3+3^2+3\right)\)

\(=2\left(3^4+3^3+3^2+3\right)\)

\(=2\left(81+27+9+3\right)\)

\(=2\left(120\right)\)

\(=240\)