Answer:
Step-by-step explanation:
Considering the geometric sequence
[tex]5,-25,\:125,\:...[/tex]
[tex]a_1=5[/tex]
As the common ratio '[tex]r[/tex]' between consecutive terms is constant.
[tex]\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{a_{n+1}}{a_n}[/tex]
[tex]r=\frac{-25}{5}=-5[/tex]
[tex]r=\frac{125}{-25}=-5[/tex]
The general term of a geometric sequence is given by the formula:
[tex]a_n=a_1\cdot \:r^{n-1}[/tex]
where [tex]a_1[/tex] is the initial term and [tex]r[/tex] the common ratio.
Putting [tex]n = 12[/tex] , [tex]r = -5[/tex] and [tex]a_1=5[/tex] in the general term of a geometric sequence to determine the 12th term of the sequence.
[tex]a_n=a_1\cdot \:r^{n-1}[/tex]
[tex]a_n=5\left(-5\right)^{n-1}[/tex]
[tex]a_{12}=5\left(-5\right)^{12-1}[/tex]
[tex]=5\left(-5^{11}\right)[/tex]
[tex]\mathrm{Remove\:parentheses}:\quad \left(-a\right)=-a[/tex]
[tex]=-5\cdot \:5^{11}[/tex]
[tex]\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}[/tex]
[tex]=-5^{1+11}[/tex] ∵ [tex]5\cdot \:5^{11}=\:5^{1+11}[/tex]
[tex]=-244140625[/tex]
Therefore,
