Question

Find an explicit rule for the geometric sequence using subscript notation. Use a calculator and round your answers to the nearest tenth if necessary.
The third term of the sequence is 170. The fifth term is 108.8.

Answer 1

Answer:

The formula for nth term is given as $a_n = (265.625)(0.8)^{(n-1)}$

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:

$a_n = a(r^{n-1})$

⇒ From above general term:  

$a_3 = a r^{2} ,\\ a_5 = ar^{4}$

⇒$170 = a r^{2} ,\\ 108.8 = ar^{4}$

Dividing both the the equations, we get:

$\frac{170}{108.8} = \frac{ar^2}{ar^4} \implies 1.5625 = \frac{1}{r^2}$

or, $r^{2} = \frac{1}{1.5625} = 0.64\\ \implies r = 0.8$

Hence, the common ratio r  = 0.8

Now, $170 = a r^2 \implies 170 = a (0.64)\\\implies a = \frac{170}{0.64} = 265.625$

⇒ a =  265.625, r = 0.8

So, the formula for nth term is given as $a_n = (265.625)(0.8)^{(n-1)}$