Question

How do I find the fifth term of a geometric sequence on a calculator?

Answer 1

Hint: According to the question we have to determine the fifth term of a geometric sequence on a calculator. So, first of all to determine the fifth term of a geometric sequence on a calculator we have to let the first term and the common ratio of the geometric sequence.
Now, to determine the required term we have to understand about the geometric sequence first term and common ratio which is as explained below:
Geometric sequence: Geometric sequence is also known as geometric sequence which is an order of the list of the numbers in which each number after the first is found by multiplying by the previous one by a fixed common number called common ratio.
Now, we have to obtain the sequence of four terms in which we have to take the first term and the common ratio as we already let.
Now, to obtain the \[{5^{th}}\] term of the geometric sequence we have to use the formula to find the \[{n^{th}}\] term of the geometric sequence which is as mentioned below:

Formula used: $ {a_n} = a{r^{n - 1}}.................(A)$
Where, a is the first term, ${a_n}$is the \[{n^{th}}\]term which we have to determine and r is the common ratio for the given geometric sequence.
Now, we just have to substitute all these values in the formula (A) as mentioned above, to determine the required term of the geometric sequence.

Complete step-by-step solution:
Step 1: First of all to determine the fifth term of a geometric sequence on a calculator we have to let the first term and the common ratio of the geometric sequence. As mentioned in the solution hint. Hence,
Let,
$ \Rightarrow $First term$ = a$
$ \Rightarrow $Common ratio$ = r$
Step 2: Now, we have to obtain the sequence of four terms in which we have to take first term and the common ratio as we already let. Hence, with the help of first term and the common ratio,
$ \Rightarrow a,ar,a{r^2},a{r^3}$………
Step 3: Now, to obtain the 5th term of the geometric sequence we have to use the formula (A) to find the \[{n^{th}}\]term of the geometric sequence which is as mentioned in the solution hint. Hence, we just have to substitute all these values in the formula (A) as mentioned above, to determine the required term of the geometric sequence.
$
   \Rightarrow {a_5} = a{r^{5 - 1}} \\
   \Rightarrow {a_5} = a{r^4}
 $

Hence, with the help of the formula (A) we have determined the required 5th term of a geometric sequence on a calculator which is $a{r^4}$.

Note: The common ratio of a geometric series may be negative, resulting in an alternating sequence. An alternating sequence will have numbers that switch back and forth between positive and negative signs.
To check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.