College

Identify the 16th term of a geometric sequence where [tex]$a_1 = 4$[/tex] and [tex]$a_8 = -8,748$[/tex].

A. [tex]-172,186,884[/tex]
B. [tex]-57,395,628[/tex]
C. [tex]57,395,628[/tex]
D. [tex]172,186,884[/tex]

Answer :

To find the 16th term of a geometric sequence where the first term [tex]\( a_1 = 4 \)[/tex] and the 8th term [tex]\( a_8 = -8,748 \)[/tex], follow these steps:

1. Identify the formula for the nth term of a geometric sequence:

[tex]\[ a_n = a_1 \times r^{(n-1)} \][/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.

2. Use the information given for the 8th term to find the common ratio [tex]\( r \)[/tex]:

[tex]\[ a_8 = a_1 \times r^{(8-1)} \][/tex]

Substituting the known values:

[tex]\[ -8,748 = 4 \times r^7 \][/tex]

To find [tex]\( r^7 \)[/tex]:

[tex]\[ r^7 = \frac{-8,748}{4} = -2,187 \][/tex]

Solve for [tex]\( r \)[/tex] by taking the 7th root of [tex]\(-2,187\)[/tex].

3. Calculate the 16th term using the formula:

[tex]\[ a_{16} = a_1 \times r^{(16-1)} = a_1 \times r^{15} \][/tex]

4. Substitute the values to find [tex]\( a_{16} \)[/tex]:

With the determined values from solving [tex]\( r^7 = -2,187 \)[/tex], compute:

[tex]\[ a_{16} = 4 \times (r^{15}) \][/tex]

5. Determine the 16th term from these calculations.

After working through the calculations, the 16th term is found to be approximately [tex]\(-57,395,628\)[/tex].

So, the 16th term of the geometric sequence is [tex]\(-57,395,628\)[/tex].