Question

How do you find a formula for the nth term of the geometric sequence: 500, 100, 20, 4, …?

Answer 1

Hint: Geometric sequence refers to the sequence which has a common ratio between the adjacent number. We will use the first number as ‘\[a\]’ and multiply the common ratio ‘\[r\]’ between the adjacent numbers using the nth term formula of Geometric progression to find the formula of the given geometric sequence.

Complete step-by-step answer:
We often use geometric sequence and geometric series interchangeably. But it is not the same. Geometric sequence can be said to be a progression of the numbers varying by the value of the common ratio ‘\[r\]’ whereas a geometric series is the sum of those numbers.
According to the question, we have to find the formula for the nth term of the given sequence.
We will start with finding the values and substituting in the formula for the nth term of a geometric sequence.
The formula is as follows:
\[{{a}_{n}}=a{{r}^{n-1}}\]
Here, ‘\[a\]’ refers to the first term or the starting number of the sequence, ‘\[r\]’ refers to the common ratio between each adjacent numbers in the geometric sequence and ‘\[{{a}_{n}}\]’ refers to the nth term in the geometric sequence.
From the sequence we have it is very evident what ‘\[a\]’ and ‘\[r\]’ are. The sequence we have is: 500, 100, 20, 4, …
So, \[a=500\]
And the common ratio can be found by dividing the second term by the first term, that is,
\[r=\dfrac{100}{500}=\dfrac{1}{5}\]
Now, we have the values of ‘\[a\]’ and ‘\[r\]’, so we will now substitute in the formula of nth term of geometric series which is, \[{{a}_{n}}=a{{r}^{n-1}}\]
We get,
\[\Rightarrow {{a}_{n}}=(500){{\left( \dfrac{1}{5} \right)}^{n-1}}\]
Therefore, the formula for the nth term in the given sequence is \[{{a}_{n}}=(500){{\left( \frac{1}{5} \right)}^{n-1}}\].

Note: Substituting the values of ‘\[a\]’ and ‘\[r\]’ in the nth term formula should be done carefully, else it will result in the wrong formula of the given geometric series. The values of ‘\[a\]’ and ‘\[r\]’ should not get reversed. The terms ‘\[{{a}_{n}}\]’ and ‘\[a\]’ are two different terms and should not be used interchangeably. ‘\[{{a}_{n}}\]’ refers to the nth term in the geometric sequence whereas ‘\[a\]’ refers to the first term or the starting number of the sequence.