Question

Divya and Miguel were asked to find an explicit formula for the sequence 80,40,20,10,..

Answer 1

Answer:

$\displaystyle a_n=80\times \left(\frac{1}{2}\right)^{n-1}$

Step-by-step explanation:

Geometric Sequence

It's a particular type of sequence where the next term is computed as the previous one times a constant number called common ration. The general formula for a geometric sequence is

$a_n=a_1\cdot r^{n-1}$

where n is the number of the term

We have to help Divya and Miguel to find an explicit formula for the sequence 80,40,20,10,..

The first term is

$a_1=80$

We need to compute the common ratio, knowing that

$a_2=40$

From the general formula

$a_n=a_1\cdot r^{n-1}$

we solve for r

$\displaystyle r=\sqrt[n-1]{\frac{a_n}{a_1}}$

$\displaystyle r=\sqrt[2-1]{\frac{40}{80}}$

$\displaystyle r=\frac{1}{2}$

Let's test further to make sure the sequence has a common ratio

$\displaystyle a_3=a_1.r^2=80\times \left(\frac{1}{2}\right)^2$

$a_3=20$

Similarly we find

$a_4=10$

The required explicit formula is

$\boxed{\displaystyle a_n=80\times \left(\frac{1}{2}\right)^{n-1}}$

Answer 2

Answer:

$a_n =80(0.5)^{n-1}$

Step-by-step explanation:

we know that

The explicit formula for a geometric sequence is  written as:

$a_n = a_1(r)^{n-1}$        

where

a_1 is the first term

r is the common ratio

we have the sequence

$80,40,20,10,..$

so

$a_1=80\\a_2=40\\a_3=20\\a_4=10$

$\frac{a_2}{a_1}=\frac{40}{80}=0.5$

$\frac{a_3}{a_2}=\frac{20}{40}=0.5$

$\frac{a_4}{a_3}=\frac{10}{20}=0.5$

therefore

$r=0.5$

substitute

$a_n =80(0.5)^{n-1}$