Question

An infinite G.P has first term $x$ and sum $5$ , then $x$ belongs to:
a.\[\mid x\mid < 10\]
b.\[ - 10 < x < 0\]
c.\[0 < x < 10\]
d.\[x > 10\]

Answer 1

Hint: First we have to define what the terms we need to solve the problem are. Sum of Infinite GP terms
If the number of terms in a GP is infinite, then the GP is called infinite GP. The formula used to find the sum to infinite terms of the given GP is:
\[{S_\infty } = \sum _{n = 1}^\infty = a{r^{n - 1}} \]
\[= \dfrac{a}{{1 - r}}\,\,\,\,\,\,\,\,; - 1 < r < 1\,\,\,\,or\,\,\,\,|r| < 1 \]
Here,
\[{S_\infty }\] = Sum of infinite geometric progression
\[a\] = First term
\[r\] = Common ratio
\[n\]= Number of terms
Rules for Solving Inequalities
Rules for solving these are similar to linear equations, however, there is only one exception that is when multiplying or dividing by a negative number, so to solve inequalities we have to follow:
1) When the addition is performed add the same number on both sides.
2) When the subtraction is performed subtract the same number on both sides.
3) By the same positive number multiply both sides.
4) By the same positive number divide both sides.
5) When multiplying with a negative number, multiply the same number on both sides and reverse the sign.
6) Divide the same negative number to both sides and reverse the sign.

Complete answer:
We are given a GP(Geometric Progression) in which the first term is \[x\]
\[x\]…….
Let the common ratio be \[r\]
According to the sum of infinite GP term
\[S = \dfrac{x}{{1 - r}} = 5\]
\[\Rightarrow x = 5(1 - r)\]
\[\Rightarrow 1 - r = \dfrac{x}{5}\]
\[\Rightarrow r = 1 - \dfrac{x}{5}\]
So the common ratio of this GP series is \[r = 1 - \dfrac{x}{5}\]
And also we know that in the case of GP
\[|r| < 1\] according to GP infinite sum formula
That also means
\[ - 1 < r < 1\]
Now placing the value of \[r\] in this equation
\[ - 1 < 1 - \dfrac{x}{5} < 1\]
Subtracting 1 from both the sides
\[ - 2 < - \dfrac{x}{5} < 0\]
Multiplying it by -1 and follow the rule for inequalities
\[0 < \dfrac{x}{5} < 2\]
\[ = 0 < x < 10\]

Hence, the correct option is \[C)\;\;\;\;0 < x < 10\]

Note:
Things that we need to give more attention: take care of signs when multiplying with numbers in inequality equations. When we multiply the same negative number on both sides sign must be reversed. Follow Rules for Solving Inequalities mentioned above.