Question

The formula for ${{n}^{th}}$ term of a geometric progression is_________.

Answer 1

To obtain the formula for ${{n}^{th}}$ term of a geometric progression we will use its definition. Firstly we will let the first term and the common difference of the geometric progression then we will find the other terms of it and according to the terms get the pattern in which the terms are found. Finally we will write the formula to find the ${{n}^{th}}$ according to the pattern and get our desired answer.

Complete step-by-step solution:
It is given to us that we have to find the formula for the ${{n}^{th}}$ term of a geometric progression.
Firstly we will define Geometric progression so,
Geometric Progression is a sequence where each succeeding term is produced by multiplying the preceding term with a fixed number also known as common ratio.
So let us take the first term of the G.P as $a$ and common ratio as $r$.
Using the above value we can say that,
Second term is:
$\begin{aligned} & {{a}_{2}}=a\times r \\\ & \therefore {{a}_{2}}=ar \\\ \end{aligned}$
Third term is:
$\begin{aligned} & {{a}_{3}}={{a}_{2}}\times r \\\ & \Rightarrow {{a}_{3}}=ar\times r \\\ & \therefore {{a}_{3}}=a{{r}^{2}} \\\ \end{aligned}$
As we can see that we have $a$ in each term and the power of the ratio is one less than the number of term so we can say that:
${{a}_{n}}=a{{r}^{n-1}}$
Hence formula for ${{n}^{th}}$ term of a geometric progression is ${{a}_{n}}=a{{r}^{n-1}}$

Note: Geometric progression is a sequence where the common ratio between the consecutive terms is the same. That means that the next term of a sequence is produced when we multiply a constant to the preceding term. The term ${{a}_{n}}$ represents the ${{n}^{th}}$ term and the term $a$ represents the first term.