Answer :
To find the 16th term of a geometric sequence, we start by using the formula for the nth term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where [tex]\( a_n \)[/tex] is the nth term, [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
We are given the first term ([tex]\( a_1 = 4 \)[/tex]) and the 8th term ([tex]\( a_8 = -8,748 \)[/tex]). We need to find the common ratio [tex]\( r \)[/tex].
### Step 1: Find the Common Ratio
We use the formula for the 8th term:
[tex]\[ a_8 = a_1 \cdot r^{(8-1)} \][/tex]
Substitute the values:
[tex]\[ -8,748 = 4 \cdot r^7 \][/tex]
To find [tex]\( r \)[/tex], solve for [tex]\( r^7 \)[/tex]:
[tex]\[ r^7 = \frac{-8,748}{4} \][/tex]
[tex]\[ r^7 = -2,187 \][/tex]
Now find the 7th root of -2,187 to find [tex]\( r \)[/tex].
### Step 2: Find the 16th Term
Once we have the common ratio, we can find the 16th term ([tex]\( a_{16} \)[/tex]) using:
[tex]\[ a_{16} = a_1 \cdot r^{(16-1)} \][/tex]
[tex]\[ a_{16} = 4 \cdot r^{15} \][/tex]
After calculating, we find that the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex].
Therefore, the 16th term of the geometric sequence is [tex]\(\boxed{57,395,628}\)[/tex].
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where [tex]\( a_n \)[/tex] is the nth term, [tex]\( a_1 \)[/tex] is the first term, and [tex]\( r \)[/tex] is the common ratio.
We are given the first term ([tex]\( a_1 = 4 \)[/tex]) and the 8th term ([tex]\( a_8 = -8,748 \)[/tex]). We need to find the common ratio [tex]\( r \)[/tex].
### Step 1: Find the Common Ratio
We use the formula for the 8th term:
[tex]\[ a_8 = a_1 \cdot r^{(8-1)} \][/tex]
Substitute the values:
[tex]\[ -8,748 = 4 \cdot r^7 \][/tex]
To find [tex]\( r \)[/tex], solve for [tex]\( r^7 \)[/tex]:
[tex]\[ r^7 = \frac{-8,748}{4} \][/tex]
[tex]\[ r^7 = -2,187 \][/tex]
Now find the 7th root of -2,187 to find [tex]\( r \)[/tex].
### Step 2: Find the 16th Term
Once we have the common ratio, we can find the 16th term ([tex]\( a_{16} \)[/tex]) using:
[tex]\[ a_{16} = a_1 \cdot r^{(16-1)} \][/tex]
[tex]\[ a_{16} = 4 \cdot r^{15} \][/tex]
After calculating, we find that the 16th term of the sequence is approximately [tex]\( 57,395,628 \)[/tex].
Therefore, the 16th term of the geometric sequence is [tex]\(\boxed{57,395,628}\)[/tex].
