Question: The sum of an infinite GP is \[x\] and common ratio \[r\] is such that \[\left| r \right| < 1\]. If ...

Question

The sum of an infinite GP is $x$ and common ratio $r$ is such that $\left| r \right| < 1$ . If the first term of the GP is $2$ , then which one of the following is correct?
A). $- 1 < x < 1$
B). $\infty < x < 1$
C). $1 < x < \infty$
D). None of the above

Answer 1

Solution

In the given question, we have been given a statement involving the use of GP, or Geometric Progression. A GP is a progression such that the quotient of any two consecutive terms is equal. We have been given all the required terms and we have to find the relation between the given terms. But the catch here is that the sum of the GP is given when the GP is given to be an infinite GP. To solve it, we are going to use the different cases of the common ratio and expand on the given terms so as to find the relation between the terms.

Complete step by step solution:
For an infinite GP with sum given as $x$ , the common ratio equal to $r$ and the first term as $2$ , we can write the sum as:
$x = \dfrac{2}{{1 - r}}$ …(i)
Here, we have been given the common ratio as:
$\left| r \right| < 1$
Case 1: $r > 0$
$\Rightarrow 0 < 1 - r < 1$
Taking reciprocal on all the sides,
$1 < \dfrac{1}{{1 - r}} < \infty$
Multiplying $2$ on all sides,
$2 < \dfrac{2}{{1 - r}} < \infty$
Using (i), we have,
$2 < x < \infty$
Case 2: $r < 0$
$1 < 1 - r < 2$
Taking reciprocal on all the sides,
$\dfrac{1}{2} < \dfrac{1}{{1 - r}} < 1$
Multiplying $2$ on all sides,
$1 < \dfrac{2}{{1 - r}} < 2$
Using (i), we have,
$1 < x < 2$
Now, taking common (union) of both the cases, we have,
$1 < x < \infty$
Hence, the correct option is C.

Note: In the given question, we were given an infinite GP, the sum of the given infinite GP, the first term of the GP and the common ratio of the GP. We had to choose the appropriate option representing the range of the sum of the given infinite GP. We solved it by considering the different cases of the common ratio and expanded on the given terms and found the relation between the required terms. So, it is very important that we have enough practice so that we can solve such questions with ease.