Question

The sum of $100$ terms of the series $0.9 + 0.09 + 0.009 + ...$ will be equal to
1. $1 - {\left( {\dfrac{1}{{10}}} \right)^{100}}$
2. $1 + {\left( {\dfrac{1}{{10}}} \right)^{106}}$
3. $1 - {\left( {\dfrac{1}{{10}}} \right)^{106}}$
4. $1 + {\left( {\dfrac{1}{{10}}} \right)^{100}}$

Answer 1

Firstly identify the first term and the common ratio of the given geometric series. Find the Sum of $n$ terms $= \dfrac{{a(1 - {r^n})}}{{1 - r}}$. Simplify the expression as much as possible and hence you get the required answer.

Complete step-by-step solution:
A geometric progression or a geometric sequence is the one, in which each term is varied by another by a common ratio. General form of a GP is
$a,ar,a{r^2},a{r^3},...,a{r^n}$
Where $a$ is the first term
$r$ is the common ratio
$a{r^n}$is the last term
If the common ratio is:
a) Negative: The result will alternate between positive and negative.
b) Greater than one: There will be an exponential growth towards infinity (positive).
c) Less than minus one: There will be an exponential growth towards infinity (positive and negative).
d) Between one and minus one:There will be an exponential decay towards zero.
e)Zero: The result will remain at zero.
Sum of $n$ terms of a GP is given by the following formula:
Sum of $n$ terms $= \dfrac{{a(1 - {r^n})}}{{1 - r}}$
where $a$ is the first term and $r$ is the common ratio of the given GP
Here in the given question we have first term, $a = 0.9$
Common ratio, $r = \dfrac{{0.09}}{{0.9}} = 0.1$
Number of terms, $n = 100$
Sum of $n$ terms $= \dfrac{{a(1 - {r^n})}}{{1 - r}}$
$= \dfrac{{0.9(1 - {{0.1}^{100}})}}{{1 - 0.1}}$
On simplification we get ,
$= \dfrac{{0.9(1 - {{0.1}^{100}})}}{{0.9}}$
On solving it further we get ,
$= 1 - {0.1^{100}}$
Hence we get ,
$= 1 - {\left( {\dfrac{1}{{10}}} \right)^{100}}$
Therefore option (1) is the correct answer.

Note: Correctly identify the first term and the common ratio of the given geometric series. Use the formula for Sum of $n$ terms of geometric expression. Simplify the expression as much as possible so as to ease the calculations. Do the calculations correctly so as to get the correct solution.