Question

What is the fifth term of the sequence \[1, - 4,16, - 64\]?

Answer 1

Hint: 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 \[b = \sqrt {ac} \].