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) \[\dfrac{121}{10}\]
b) \[\dfrac{441}{100}\]
c) \[100\]
d) \[110\]

Answer 1

Hint:In the given question, we have been asked to find the value of ‘k’ given that \[{{\left( 10\right)}^{9}}+2{{\left( 11 \right)}^{1}}{{\left( 10 \right)}^{8}}+3{{\left( 11 \right)}^{2}}{{\left( 10 \right)}^{7}}+--+10{{\left( 11 \right)}^{9}}=k{{\left( 10 \right)}^{9}}\]. In order to solve the question, we just need to simplify the question using some basic mathematical operations such as addition, subtraction, multiplication and division. We have to use the formula of sum of ‘n’ terms of GP using the formula of; In GP, sum of n terms \[=\dfrac{a\left( {{r}^{n}}-1 \right)}{r-1},\ where\ r>1\]. On further simplification, we will get the required solution.

Complete step by step solution:
We have given,
\[\Rightarrow {{\left( 10 \right)}^{9}}+2{{\left( 11 \right)}^{1}}{{\left( 10 \right)}^{8}}+3{{\left( 11\right)}^{2}}{{\left( 10 \right)}^{7}}+--+10{{\left( 11 \right)}^{9}}=k{{\left( 10 \right)}^{9}}\]
Dividing each term of both the sides of the equation by \[{{\left( 10 \right)}^{9}}\], we obtain
\[\Rightarrow \dfrac{{{\left( 10 \right)}^{9}}}{{{\left( 10 \right)}^{9}}}+\dfrac{2{{\left( 11
\right)}^{1}}{{\left( 10 \right)}^{8}}}{{{\left( 10 \right)}^{9}}}+\dfrac{3{{\left( 11 \right)}^{2}}{{\left( 10
\right)}^{7}}}{{{\left( 10 \right)}^{9}}}+--+\dfrac{10{{\left( 11 \right)}^{9}}}{{{\left( 10
\right)}^{9}}}=\dfrac{k{{\left( 10 \right)}^{9}}}{{{\left( 10 \right)}^{9}}}\]
On simplifying the above, we get
\[\Rightarrow 1+2\left( \dfrac{11}{10} \right)+3{{\left( \dfrac{11}{10} \right)}^{2}}+---+10{{\left(
\dfrac{11}{10} \right)}^{9}}=k\]------ (1)
Multiplying each term of both the sides of the equation by\[\dfrac{11}{10}\], we obtain
\[\Rightarrow 1\left( \dfrac{11}{10} \right)+2{{\left( \dfrac{11}{10} \right)}^{2}}+3{{\left( \dfrac{11}{10}
\right)}^{3}}+---+10{{\left( \dfrac{11}{10} \right)}^{10}}=k\left( \dfrac{11}{10} \right)\]-------- (2)
Subtracting equation (2) from equation (1), we obtain
\[\Rightarrow 1+\left( \dfrac{11}{10} \right)+{{\left( \dfrac{11}{10} \right)}^{2}}+--+{{\left(
\dfrac{11}{10} \right)}^{9}}-10{{\left( \dfrac{11}{10} \right)}^{10}}=k\left( 1-\dfrac{11}{10} \right)\]
We can see that all the terms are in GP, except the last term of the sequence
Thus,
\[a={{10}^{9}}\ and\ r=\dfrac{11}{10}\],
In GP, sum of n terms \[=\dfrac{a\left( {{r}^{n}}-1 \right)}{r-1},\ where\ r>1\]
On further simplification, we obtain
\[\Rightarrow \dfrac{1\left[ {{\left( \dfrac{11}{10} \right)}^{10}}-1 \right]}{\left( \dfrac{11}{10}-1 \right)}-
10{{\left( \dfrac{11}{10} \right)}^{10}}=k\left( \dfrac{10-11}{10} \right)\]
Solving for the value of k, we get
\[\Rightarrow 10\left[ 10{{\left( \dfrac{11}{10} \right)}^{10}}-10-10{{\left( \dfrac{11}{10} \right)}^{10}}
\right]=-k\]
\[\Rightarrow -100=-k\]
On cancelling out the negative sign from both the sides of the equation, we get
\[\Rightarrow k=100\]
Therefore, the value of k is equals to 100.
Hence, the option (c ) is the correct answer.
Formula used:
In GP, sum of n terms = \[=\dfrac{a\left( {{r}^{n}}-1 \right)}{r-1},\ where\ r>1\]

Note: To solve these types of question where we required to add the terms given in a sequence, sometimes we just need to use the formula of arithmetic series i.e. AP and GP. Students should carefully write each and every term explicitly to avoid any calculation mistakes as these types of questions require a lot of calculation, so we should avoid making errors.