Question

What is the common ratio in GP?

Answer 1

Hint: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers as it follows a pattern.

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 denoted as $$r$$. 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 and $$a{r}^n$$ is the last term.
Explanation of common ratio of a GP:
Consider a GP $$a,ar,a{r}^2,a{r}^3,...,a{r}^n$$ where
First term $$=a$$
Second term $$=ar$$
Third term $$=a{r}^2$$
Nth term $$=a{r}^{n-1}$$
Therefore common ratio $$={\displaystyle \frac{anyterm}{precedingterm}}$$
 $$={\displaystyle \frac{\text{Third term}}{\text{Second term}}}$$
$$={\displaystyle \frac{a{r}^2}{ar}}=r$$
If the common ratio is:
a) Negative: The result will alternate between positive and negative.
b) Greater than$$1$$: There will be an exponential growth towards infinity (positive).
c) Less than $$-1$$: There will be an exponential growth towards infinity (positive and negative).
d) Between $$1$$ and $$-1$$: 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 $$={\displaystyle \frac{a(1-r^n)}{1-r}}$$
Where $$a$$ is the first term and $$r$$ is the common ratio of the given GP.

Note: This is a theoretical type of question so we need to have proper knowledge of the concept of Geometric Progression and its properties. We should know the general form of the geometric progression in order to find the solution to the given question.