Question

How do you write an nth term rule for $ r = 6 $ and $ {a_3} = 72 $ ?

Answer 1

Hint: Sequence is basically a set of things that are in any order. Geometric sequence is a sequence where the ratio between each successive pair of terms is the same and it is abbreviated as GP. Next term of any sequence can be obtained by multiplying a constant number to the term before it. That constant number which is multiplied is known as the common ratio ( $ r $ ). Since, all the geometric sequences follow the same pattern, we can write the same rule for finding the nth term for the sequence.

Complete step by step solution:
We are given,
 $ r = 6 $
 $ {a_3} = 72 $
To find the nth term,
 $ {a_n} = a{r^{n - 1}} $
To find nth term for any expression we need its first term and common ratio, so to find the nth rule for the following question, we need to find $ a $
 $ \Rightarrow 72 = a \times {6^{3 - 1}} $
 $ \Rightarrow 72 = a \times {6^2} $
 $ \Rightarrow 72 = 36a $
 $ \Rightarrow a = \dfrac{{72}}{{36}} $
 $ \Rightarrow a = 2 $
The nth rule would be,
 $ \Rightarrow {a_n} = 2 \times {6^{n - 1}} $
This is the required answer.
So, the correct answer is “${a_n} = 2 \times {6^{n - 1}} $ ”.

Note: Common ratio of any sequence can be obtained by dividing any of the two terms i.e. subtracting the latter term from prior.
\[r = \dfrac{{{a_n}}}{{{a_{n - 1}}}}\]
Also, Geometric Sequence can be both finite and infinite. The behavior of GP depends on the nature of common ratio.
If the common ratio is a positive number, the sequence will progress towards infinity.
If the common ratio is a negative number, the sequence will regress towards negative infinity.