Question

Find the value of \[k\] if \[\;{\left( {10} \right)^9}\; + {\rm{ }}2{\left( {11} \right)^1}\;{\left( {10} \right)^{8\;}} + {\rm{ }}3{\left( {11} \right)^2}\;{\left( {10} \right)^7}\; + {\rm{ }} \ldots + {\rm{ }}10{\left( {11} \right)^9}\; = {\rm{ }}k{\left( {10} \right)^9}\]
A. \[100\]
B. \[110\]
C. \[\dfrac{{121}}{{100}}\]
D. \[\dfrac{{441}}{{100}}\]

Answer 1

Hint: We have given \[\;{\left( {10} \right)^9}\; + {\rm{ }}2{\left( {11} \right)^1}\;{\left( {10} \right)^{8\;}} + {\rm{ }}3{\left( {11} \right)^2}\;{\left( {10} \right)^7}\; + {\rm{ }} \ldots + {\rm{ }}10{\left( {11} \right)^9}\; = {\rm{ }}k{\left( {10} \right)^9}\] and we have to find the value of \[k\] in the given equation. We will use the concept simple mathematics and sum of geometric series in the given question. So firstly we will make the equation simpler by arithmetic’s and then find the value of \[k\].
Formula used: The formula of sum of geometric progression will be useful for this question
\[{S_n} = \dfrac{{a({r^n} - 1)}}{{r - 1}}\] where \[r \ne 1\]
Complete step-by-step solution:
Firstly we will cancel \[{10^9}\] from both sides
\[k{\rm{ }} = {\rm{ }}1 + {\rm{ }}2\left( {\dfrac{{11}}{{10}}} \right){\rm{ }} + {\rm{ }}3{\left( {\dfrac{{11}}{{10}}} \right)^2}\; + {\rm{ }} \ldots .10{\left( {\dfrac{{11}}{{10}}} \right)^9}\] … \[(1)\]
Now, we will multiply L.H.S and R.H.S by \[\dfrac{{11}}{{10}}\]
\[\left( {\dfrac{{11}}{{10}}} \right)k{\rm{ }} = {\rm{ }}\left( {\dfrac{{11}}{{10}}} \right){\rm{ }} + {\rm{ }}2{\left( {\dfrac{{11}}{{10}}} \right)^2}\; + {\rm{ }} \ldots . + {\rm{ }}9{\left( {\dfrac{{11}}{{10}}} \right)^{9\;}} + {\rm{ }}10{\left( {\dfrac{{11}}{{10}}} \right)^{10}}\] … \[(2)\]
Now, we will subtract equation \[(2)\] from equation \[(1)\]
\[\begin{array}{l}k\left( {1{\rm{ }}-{\rm{ }}\dfrac{{11}}{{10}}} \right){\rm{ }} = {\rm{ }}1 + {\rm{ }}\left( {\dfrac{{11}}{{10}}} \right){\rm{ }} + {\rm{ }}{\left( {\dfrac{{11}}{{10}}} \right)^2}\; + {\rm{ }} \ldots .{\left( {\dfrac{{11}}{{10}}} \right)^{9\;}}-{\rm{ }}10{\left( {\dfrac{{11}}{{10}}} \right)^{10}}\\k\dfrac{{\left( {10{\rm{ }}-{\rm{ }}11} \right)}}{{10}}{\rm{ }} = {\rm{ }}1\left[ {\dfrac{{{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}}\;-{\rm{ }}1}}{{\left( {\dfrac{{11}}{{10}}} \right){\rm{ }} - 1}}} \right]{\rm{ }}-{\rm{ }}10{\left( {\dfrac{{11}}{{10}}} \right)^{10}}\end{array}\]
Using the formula for sum of geometric series
\[\begin{array}{l} - k{\rm{ }} = {\rm{ }}10\left[ {10{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}}\;-{\rm{ }}10{\rm{ }}-{\rm{ }}10{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}}} \right]\\ - k = 10[ - 10]\\ - k = - 100\\k = 100\end{array}\]
Therefore, the option (A) is correct.
Additional Information:A geometric progression, often referred to as a geometric sequence, is a series of non-zero values where each term following the first is obtained by multiplying the preceding value by a constant, non-zero number known as the common ratio. The terms that follow from multiplying or dividing each GP term by a non-zero value also belong to GP. The resulting terms are also in GP if all of the terms in GP are elevated to the same powers. The GP's standard form is\[a,{\rm{ }}ar,{\rm{ }}a{r^2}{\rm{ }},{\rm{ }}a{r^3}...\]
Note: Students generally make the mistake in calculation and applying the formula of sum of geometric progression. To avoid this mistake, the formula of sum of GP should be clear and calculation part should be strong. The fraction’s part calculation should be done carefully to get correct answer.