Question

The sum of 100 terms of the series $0.9 + 0.09 + 0.009$ will be?
A. $1 - {\left( {\dfrac{1}{{10}}} \right)^{100}}$
B. $1 + {\left( {\dfrac{1}{{10}}} \right)^{106}}$
C. $1 - {\left( {\dfrac{1}{{10}}} \right)^{106}}$
D. $1 + {\left( {\dfrac{1}{{10}}} \right)^{100}}$

Answer 1

We can find the total sum of series by using progression. For this we have to observe which progression is series in whether geometric progression, arithmetic progression or harmonic progression. And when we find that then we will apply the sum formula to find the sum of 100 terms of a given series.

Complete step-by-step answer:
First of all we will write the given equation in fraction.
$\Rightarrow 0.9 + 0.09 + 0.009$
$\Rightarrow \dfrac{9}{{10}} + \dfrac{9}{{100}} + \dfrac{9}{{1000}}$ As we can see that terms in the series are increasing with the factor of $\dfrac{1}{{10}}$ so we can say that this series is of geometric progression.
The general formula of geometric progression is $a,a{r^2},a{r^3}$ and so on. Here $'a'$ is first term and $'r'$ is the common ratio.
${n^{th}}$ Term of geometric progression is given by ${a_n} = a{r^{n - 1}}$
Common ratio ${r_n} = \dfrac{{{a_n}}}{{{a_{n - 1}}}}$
Sum of first n terms of G.P is given by ${S_n} = \dfrac{{{a_1}(1 - {r^n})}}{{1 - r}}$
Now we will find common ratio,
$\Rightarrow {r_n} = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{9/100}}{{9/10}}$
$\Rightarrow {r_n} = \dfrac{1}{{10}} = 0.1$
Now we will find sum of 100 terms by applying formula ${S_n} = \dfrac{{{a_1}(1 - {r^n})}}{{1 - r}}$
$\Rightarrow {S_{100}} = \dfrac{{0.9(1 - {{0.1}^{100}})}}{{1 - 0.1}}$
$\Rightarrow {S_{100}} = \dfrac{{0.9(1 - {{0.1}^{100}})}}{{0.9}}$
Now we will cancel 0.9 from the numerator and denominator.
$\Rightarrow {S_{100}} = 1 - {\left( {\dfrac{1}{{10}}} \right)^{100}}$
Therefore sum of 100 terms of the series $0.9 + 0.09 + 0.009$ is $1 - {\left( {\dfrac{1}{{10}}} \right)^{100}}$

So, the correct answer is “Option A”.

Note: Students can be confused while determining type of series whether it is a G.P, A.P or H.P so here below definition of them is being mentioned which states that arithmetic progression is a sequence of numbers such that the difference of any two successive numbers is a constant and that constant value is known as common difference, while harmonic progression is a sequence of real numbers formed by reciprocal of arithmetic progression.