Answer:
The missing term is:
[tex]a_2=14[/tex]
Step-by-step explanation:
Considering the geometric sequence
[tex]2,\:?,\:98,\:...[/tex]
Here,
[tex]a_1=2[/tex]
[tex]a_3=98[/tex]
[tex]a_2=?[/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.
as
[tex]a_3=a_1\cdot \:r^{3-1}[/tex]
[tex]98=2\cdot \:r^{3-1}[/tex] ∵ [tex]a_1=2[/tex]
[tex]\mathrm{Switch\:sides}[/tex]
[tex]2r^{3-1}=98[/tex]
[tex]\mathrm{Divide\:both\:sides\:by\:}2[/tex]
[tex]\frac{2r^{3-1}}{2}=\frac{98}{2}[/tex]
[tex]r^{3-1}=49[/tex]
[tex]r^2=49[/tex]
[tex]\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}[/tex]
[tex]r=\sqrt{49},\:r=-\sqrt{49}[/tex]
[tex]r=7,\:r=-7[/tex]
Taking positive value of [tex]r = 7[/tex]
[tex]a_2=a_1\cdot \:r^{n-1}[/tex]
putting [tex]r = 7[/tex]
[tex]a_2=2\cdot \:7^{2-1}[/tex]
[tex]a_2=2\cdot \:7[/tex]
[tex]a_2=14[/tex]
So, the sequence becomes [tex]2,\:14,\:98,\:...[/tex]
Therefore, the missing term is:
[tex]a_2=14[/tex]
