If \((10)^9\) + \(2(11)^1(10)^8\) + \(3(11)^2(10)^7\) + ...... + \(10(11)^9\) = \(K(10)^9\), then k is equal to

Solution :

\(K(10)^9\) = \((10)^9\) + \(2(11)^1(10)^8\) + \(3(11)^2(10)^7\) + …… + \(10(11)^9\)

K = 1 + 2\(({11\over 10})\) + 3\(({11\over 10})^2\) + ….. + 10\(({11\over 10})^9\) ……(i)

\(({11\over 10})\)K = 1\(({11\over 10})\) + 2\(({11\over 10})^2\) + 3\(({11\over 10})^3\) + ….. + 10\(({11\over 10})^{10}\) …..(ii)

On subtracting equation (ii) from (i), we get

K\((1 – {11\over 10})\) = 1 + \(({11\over 10})\) + \(({11\over 10})^2\) + …. + \(({11\over 10})^9\) – 10\(({11\over 10})^{10}\)

\(\implies\) K\(({10 – 11\over 10})\) = \(1[({11\over 10})^{10} – 1]\over ({11\over 10} – 1)\) – 10\(({11\over 10})^{10}\)

\(\implies\) – K = 10[10\(({11\over 10})^{10}\) – 10 – 10\(({11\over 10})^{10}\) ]

\(\implies\) K = 100