Answer :
Answer:
The formula for nth term is given as [tex]a_n = (265.625)(0.8)^{(n-1)}[/tex]
Step-by-step explanation:
Here, 3rd term in the geometric sequence = 170
The 5th term in the sequence = 108.8
Let the first term in the sequence = a
Let the common ratio of the sequence = r
Now by the Geometric Sequence:
[tex]a_n = a(r^{n-1})[/tex]
⇒ From above general term:
[tex]a_3 = a r^{2} ,\\ a_5 = ar^{4}[/tex]
⇒[tex]170 = a r^{2} ,\\ 108.8 = ar^{4}[/tex]
Dividing both the the equations, we get:
[tex]\frac{170}{108.8} = \frac{ar^2}{ar^4} \implies 1.5625 = \frac{1}{r^2}[/tex]
or, [tex]r^{2} = \frac{1}{1.5625} = 0.64\\ \implies r = 0.8[/tex]
Hence, the common ratio r = 0.8
Now, [tex]170 = a r^2 \implies 170 = a (0.64)\\\implies a = \frac{170}{0.64} = 265.625[/tex]
⇒ a = 265.625, r = 0.8
So, the formula for nth term is given as [tex]a_n = (265.625)(0.8)^{(n-1)}[/tex]