Question

What is the fifth term of the sequence $1, - 4,16, - 64$?

Answer 1

We will first find the type of progression and then calculate its ${n^{th}}term$ to solve this problem then we will substitute n=5 to get the required answer.
We will know about the arithmetic progression and also about the geometric progression and ways to verify whether a given sequence is an Arithmetic progression or a Geometric progression.

Complete step by step answer:
In mathematics and mainly in sequences and progressions concepts, there are mainly two types of progressions.
Those are Arithmetic Progression (AP) and Geometric progression (GP).
If a term is continuously added to its previous term, then that sequence is called an AP.
Example of an AP is $4,7,10,13,.....$
Here, 3 is getting added to its previous term continuously and we are getting a new term.
If a term is continuously multiplied to its previous term, then that is a GP.
Example of a GP is $1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},......$
Here, $\dfrac{1}{2}$ is getting multiplied to its previous term continuously and we are getting a new term.
Consider a sequence $a,b,c$
This sequence is
(1) Arithmetic Progression if there exist a common difference i.e., if $b - a = c - b = d({\text{constant)}}$
(2) Geometric Progression if there exist a common ratio i.e., if $\dfrac{b}{a} = \dfrac{c}{b} = r({\text{constant)}}$
So, now in the given sequence, $1, - 4,16, - 64$, there is no constant common difference, so this sequence is not an AP.
And $\dfrac{{ - 4}}{1} = \dfrac{{16}}{{ - 4}} = \dfrac{{ - 64}}{{16}} = - 4({\text{constant)}}$
So, common ratio $r = - 4$
So, it is a GP.
Generally, a GP is written as $a,ar,a{r^2},a{r^3},.....$
And ${n^{th}}$ term is given by ${a_n} = a{r^{n - 1}}$
So, here, the fifth term is ${a_5} = a{r^{5 - 1}} = 1{( - 4)^4}$
$\Rightarrow {a_5} = 256$
So, the fifth term is 256.

Note:
The first term of an arithmetic progression or a geometric progression can be a negative number also. But the first term of a GP should be a non-zero value. And the common difference and common ratio can be a negative or positive or a fractional value too. (Common ratio should not be a zero)
If the terms $a,b,c$ are in AP, then $b = \dfrac{{a + c}}{2}$
If the terms $a,b,c$ are in GP, then $b = \sqrt {ac}$.