Answer :
We start with the formula for the [tex]\( n \)[/tex]th term of a geometric sequence:
[tex]$$
a_n = a_1 \cdot r^{\,n-1},
$$[/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
Given that
[tex]$$
a_1 = 4 \quad \text{and} \quad a_8 = -8748,
$$[/tex]
we have
[tex]$$
a_8 = 4 \cdot r^7 = -8748.
$$[/tex]
To find [tex]\( r^7 \)[/tex], we divide both sides of the equation by 4:
[tex]$$
r^7 = \frac{-8748}{4} = -2187.
$$[/tex]
Notice that [tex]\( 2187 = 3^7 \)[/tex]. Therefore, we can write
[tex]$$
r^7 = -3^7.
$$[/tex]
Taking the 7th root of both sides, we obtain
[tex]$$
r = -3.
$$[/tex]
Now, to find the 16th term [tex]\( a_{16} \)[/tex], we use
[tex]$$
a_{16} = a_1 \cdot r^{15}.
$$[/tex]
Substitute the known values:
[tex]$$
a_{16} = 4 \cdot (-3)^{15}.
$$[/tex]
Calculating [tex]\( (-3)^{15} \)[/tex] gives a negative number since an odd power of a negative number remains negative. The numerical result is:
[tex]$$
(-3)^{15} = -14348907.
$$[/tex]
Thus, we have
[tex]$$
a_{16} = 4 \cdot (-14348907) = -57395628.
$$[/tex]
Hence, the 16th term of the geometric sequence is
[tex]$$
\boxed{-57395628}.
$$[/tex]
[tex]$$
a_n = a_1 \cdot r^{\,n-1},
$$[/tex]
where [tex]\( a_1 \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
Given that
[tex]$$
a_1 = 4 \quad \text{and} \quad a_8 = -8748,
$$[/tex]
we have
[tex]$$
a_8 = 4 \cdot r^7 = -8748.
$$[/tex]
To find [tex]\( r^7 \)[/tex], we divide both sides of the equation by 4:
[tex]$$
r^7 = \frac{-8748}{4} = -2187.
$$[/tex]
Notice that [tex]\( 2187 = 3^7 \)[/tex]. Therefore, we can write
[tex]$$
r^7 = -3^7.
$$[/tex]
Taking the 7th root of both sides, we obtain
[tex]$$
r = -3.
$$[/tex]
Now, to find the 16th term [tex]\( a_{16} \)[/tex], we use
[tex]$$
a_{16} = a_1 \cdot r^{15}.
$$[/tex]
Substitute the known values:
[tex]$$
a_{16} = 4 \cdot (-3)^{15}.
$$[/tex]
Calculating [tex]\( (-3)^{15} \)[/tex] gives a negative number since an odd power of a negative number remains negative. The numerical result is:
[tex]$$
(-3)^{15} = -14348907.
$$[/tex]
Thus, we have
[tex]$$
a_{16} = 4 \cdot (-14348907) = -57395628.
$$[/tex]
Hence, the 16th term of the geometric sequence is
[tex]$$
\boxed{-57395628}.
$$[/tex]
