Given:
The sequence is:
[tex]6, -24, 96, -384,...[/tex]
To find:
The explicit formula for the given sequence and then find the 16th term.
Solution:
We have,
[tex]6, -24, 96, -384,...[/tex]
The ratio between two consecutive terms are:
[tex]\dfrac{-24}{6}=-4[/tex]
[tex]\dfrac{96}{-24}=-4[/tex]
[tex]\dfrac{-384}{96}=-4[/tex]
The given sequence has a common ratio. So, the given sequence is a geometric sequence with first term 6 and common ratio [tex]-4[/tex].
The explicit formula of a geometric sequence is:
[tex]a_n=ar^{n-1}[/tex]
Where, a is the first term and r is the common ratio.
Putting [tex]a=6, r=-4[/tex] in the above formula, we get
[tex]a_n=6(-4)^{n-1}[/tex]
We need to find the 16th term. So, put [tex]n=16[/tex] in the above formula.
[tex]a_n=6(-4)^{16-1}[/tex]
[tex]a_n=6(-4)^{15}[/tex]
[tex]a_n=6(-1073741824)[/tex]
[tex]a_n=-6442450944[/tex]
Therefore, the explicit formula for the given sequence is [tex]a_n=6(-4)^{n-1}[/tex] and 16th term of the given sequence is [tex]-6.442450944[/tex].