Question

What is the ${{n}^{th}}$ term of the geometric sequence$360,\text{ }180,\text{ }90,\text{ }45$ ?

Answer 1

For the type of question we need to find out the common ratio, as in geometric progression each term differs by a common ratio. As in general form geometric progression occurs in a way $a,ar,a{{r}^{2}}...........a{{r}^{n-1}}$ where a is the 1st term and $a{{r}^{n-1}}$is the ${{n}^{th}}$ term. And to find the common ratio of a geometric sequence is just to divide the ${{2}^{nd}}$ term by ${{1}^{st}}$ term.

Complete step by step solution:
In order to know the ${{n}^{th}}$ term of our question we just need to know the 1st term and common ratio and put their value in the general ${{n}^{th}}$terms of geometric expression.
Since the expression given to us $360,\text{ }180,\text{ }90,\text{ }45$in which the first term is 360 and 2nd term is 180. So to find the common ratio divide the ${{2}^{nd}}$ term by ${{1}^{st}}$ term as a way to find common ratio in geometric progression.
So the common ratio of the given sequence is$\dfrac{180}{360}=\dfrac{1}{2}$ .
So $\dfrac{1}{2}$is the common ratio, whose 1st term i.e. a is 360.
So as we know the general ${{n}^{th}}$term of the sequence is$a{{r}^{n-1}}$.
The general ${{n}^{th}}$term of our sequence is $360\times {{\left( \dfrac{1}{2} \right)}^{n-1}}$
Hence $360\times {{\left( \dfrac{1}{2} \right)}^{n-1}}$is the ${{n}^{th}}$ term of the sequence.

Note: By just finding the common ratio and 1st term in the geometric sequence we can easily find any of the terms in the sequence. The common ratio can be greater than 1 or less than 1. As in our case, it is$\dfrac{1}{2}$, less than 1.