Question

If ${\left( {10} \right)^9} + 2{\left( {11} \right)^1}{\left( {10} \right)^8} + 3{\left( {11} \right)^2}{\left( {10} \right)^7} + ..... + 10{\left( {11} \right)^9} = k{\left( {10} \right)^9}$ , then k is equal to
A. $100$
B. $110$
C. $\dfrac{{121}}{{100}}$
D. $\dfrac{{441}}{{100}}$

Answer 1

Hint: Simplification is the process of writing the given algebraic expression effectively and most comfortably to understand without affecting the original expression. Moreover, various steps are involved to simplify an expression.
Here, in this question, we need to apply the BODMAS rule (i.e.) we need to calculate the brackets first and then orders, then division or multiplication, and finally we need to add or subtract.
Formula to be used:
a) $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
b) $\dfrac{1}{{{a^m}}} = {a^{ - m}}$
c)${a^m}{a^n} = {a^{m + n}}$
d) The formula to calculate the sum of the geometric series is as follows.
$1 + x + {x^2} + .... + {x^n} = \dfrac{{1 - {x^{n + 1}}}}{{1 - x}}$ where$x < 1$

Complete step by step answer:
The given equation is ${\left( {10} \right)^9} + 2{\left( {11} \right)^1}{\left( {10} \right)^8} + 3{\left( {11} \right)^2}{\left( {10} \right)^7} + ..... + 10{\left( {11} \right)^9} = k{\left( {10} \right)^9}$
To find: The value of k.
${\left( {10} \right)^9} + 2{\left( {11} \right)^1}{\left( {10} \right)^8} + 3{\left( {11} \right)^2}{\left( {10} \right)^7} + ..... + 10{\left( {11} \right)^9} = k{\left( {10} \right)^9}$
Let us divide the term ${\left( {10} \right)^9}$ into both sides.
That is $\dfrac{{{{\left( {10} \right)}^9}}}{{{{\left( {10} \right)}^9}}} + \dfrac{{2{{\left( {11} \right)}^1}{{\left( {10} \right)}^8}}}{{{{\left( {10} \right)}^9}}} + {\dfrac{{3{{\left( {11} \right)}^2}\left( {10} \right)}}{{{{\left( {10} \right)}^9}}}^7} + ..... + \dfrac{{10{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} = k\dfrac{{{{\left( {10} \right)}^9}}}{{{{\left( {10} \right)}^9}}}$
We shall apply $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$here.
$ \Rightarrow 1 + 2{\left( {11} \right)^1}{\left( {10} \right)^{8 - 9}} + 3{\left( {11} \right)^2}{\left( {10} \right)^{7 - 9}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} = k{10^{9 - 9}}$
$ \Rightarrow 1 + 2{\left( {11} \right)^1}{\left( {10} \right)^{ - 1}} + 3{\left( {11} \right)^2}{\left( {10} \right)^{ - 2}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} = k{10^0}$
Now, we shall apply $\dfrac{1}{{{a^m}}} = {a^{ - m}}$.
$ \Rightarrow 1 + 2{\left( {11} \right)^1}\dfrac{1}{{\left( {10} \right)}} + 3{\left( {11} \right)^2}\dfrac{1}{{{{\left( {10} \right)}^2}}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} = k$ ………….$\left( 1 \right)$
Now, we shall multiply the term $\dfrac{{11}}{{10}}$ on both sides.
 \[\dfrac{{11}}{{10}} + 2{\left( {11} \right)^1}\dfrac{1}{{\left( {10} \right)}}\dfrac{{11}}{{10}} + 3{\left( {11} \right)^2}\dfrac{1}{{{{\left( {10} \right)}^2}}}\dfrac{{11}}{{10}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}}\dfrac{{11}}{{10}} = k\dfrac{{11}}{{10}}\]
Here we shall apply ${a^m}{a^n} = {a^{m + n}}$.
$ \Rightarrow \dfrac{{11}}{{10}} + 2{\left( {11} \right)^{1 + 1}}\dfrac{1}{{{{\left( {10} \right)}^{1 + 1}}}} + 3{\left( {11} \right)^{2 + 1}}\dfrac{1}{{{{\left( {10} \right)}^{2 + 1}}}} + ..... + 10\dfrac{{{{\left( {11} \right)}^{9 + 1}}}}{{{{\left( {10} \right)}^{9 + 1}}}} = k\dfrac{{11}}{{10}}$
$ \Rightarrow \dfrac{{11}}{{10}} + 2\dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} + 3\dfrac{{{{\left( {11} \right)}^3}}}{{{{\left( {10} \right)}^3}}} + ..... + 10\dfrac{{{{\left( {11} \right)}^{10}}}}{{{{10}^{10}}}} = k\dfrac{{11}}{{10}}$ …………….$\left( 2 \right)$
Now, we need to subtract $\left( 1 \right)$ from $\left( 2 \right)$
$ \Rightarrow 1 + 2\dfrac{{{{\left( {11} \right)}^1}}}{{\left( {10} \right)}} + 3\dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} - \dfrac{{11}}{{10}} - 2\dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} - 3\dfrac{{{{\left( {11} \right)}^3}}}{{{{\left( {10} \right)}^3}}} - ..... - 10\dfrac{{{{\left( {11} \right)}^{10}}}}{{{{10}^{10}}}} = k - k\dfrac{{11}}{{10}}$
$ \Rightarrow 1 + 2\dfrac{{{{\left( {11} \right)}^1}}}{{\left( {10} \right)}} - \dfrac{{11}}{{10}} + 3\dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} - 2\dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} + ..... + 10\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} - 9\dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} - 10\dfrac{{{{\left( {11} \right)}^{10}}}}{{{{10}^{10}}}} = k\left( {1 - \dfrac{{11}}{{10}}} \right)$
$ \Rightarrow 1 + \dfrac{{{{\left( {11} \right)}^1}}}{{\left( {10} \right)}} + \dfrac{{{{\left( {11} \right)}^2}}}{{{{\left( {10} \right)}^2}}} + ..... + \dfrac{{{{\left( {11} \right)}^9}}}{{{{\left( {10} \right)}^9}}} - 10\dfrac{{{{\left( {11} \right)}^{10}}}}{{{{10}^{10}}}} = k\left( {1 - \dfrac{{11}}{{10}}} \right)$
\[ \Rightarrow 1 + {\left( {\dfrac{{11}}{{10}}} \right)^1} + {\left( {\dfrac{{11}}{{10}}} \right)^2} + ..... + {\left( {\dfrac{{11}}{{10}}} \right)^9} - 10{\left( {\dfrac{{11}}{{10}}} \right)^{10}} = k\left( {\dfrac{{10 - 11}}{{10}}} \right)\]
Now, use the formula $1 + x + {x^2} + .... + {x^n} = \dfrac{{1 - {x^{n + 1}}}}{{1 - x}}$
\[ \Rightarrow k\left( {\dfrac{{ - 1}}{{10}}} \right) = \dfrac{{1\left[ {{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}} - 1} \right]}}{{\dfrac{{11}}{{10}} - 1}} - 10{\left( {\dfrac{{11}}{{10}}} \right)^{10}}\]
\[ \Rightarrow k\left( {\dfrac{{ - 1}}{{10}}} \right) = 10\left[ {{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}} - 1} \right] - 10{\left( {\dfrac{{11}}{{10}}} \right)^{10}}\]
\[ \Rightarrow - k = 10\left[ {10{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}} - 10 - 10{{\left( {\dfrac{{11}}{{10}}} \right)}^{10}}} \right]\]
\[ \Rightarrow - k = 10\left[ { - 10} \right]\]
\[ \Rightarrow k = 100\]

So, the correct answer is “Option A”.

Note: Simplification of an expression is the process of changing the expression effectively without changing the meaning of an expression.
Here, in this question, we need to apply the BODMAS rule (i.e.) we need to calculate the brackets first and then orders, then division or multiplication, and finally we need to add or subtract.
Moreover, various steps are involved to simplify an expression. Some of the steps are listed below:
If the given expression contains like terms, we need to combine them.
Example: $3x + 2x + 4 = 5x + 4$
We need to split an expression into factors (i.e) the process of finding the factors for the given expression.
Example: ${x^2} + 4x + 3 = (x + 3)(x + 1)$
We need to expand an algebraic expression (i.e) we have to remove the respective brackets of an expression.
Example: $3(a + b) = 3a + 3b$.
We need to cancel out the common terms in an expression.