kyleryoung0602

kyleryoung0602

27.06.2020 •

Mathematics

Researchers measured the data speeds for a particular smartphone carrier at 50 randomly selected airports. The highest speed measured was 74.6 megabits per second (Mbps). The complete set of 50 data speeds has a mean of 17.05 Mbps and a standard deviation of 20.54 Mbps. What is the z-score of the highest speed measured (74.6 Mbps)

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Ответ:

rainyfallaw

02.07.2022

$b_n = \frac{2}{27} (9^n)$

Step-by-step explanation:

Given

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Required

An explicit rule for $b_1,\ b_2,\ b_3,\ \dots$

Where $b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

We have:

$a_1=\frac{2}{3},\ a_n=3a_{n-1}.$

Calculate a2

$a_2 = 3a_{2-1}$

a_2 = 3a_1

$a_2 = 3 * \frac{2}{3}$

a_2 = 2

Calculate a3

$a_3 = 3a_{3-1}$

a_3 = 3a_2

a_3 = 3 *2

a_3 = 6

Calculate a4

$a_4 = 3a_{4-1}$

a_4 = 3a_3

a_4 = 3*6

a_4 = 18

Calculate a5

$a_5 = 3a_{5-1}$

a_5 = 3a_4

a_5 = 3*18

a_5 = 54

So:

$b_1=a_1,\ b_2=a_3,\ b_3=a_5,\ \dots$

$a_1=\frac{2}{3}$ a_3 = 6 a_5 = 54

The above sequence form a geometric sequence.

Calculate common ratio (r)

$r = \frac{b_3}{b_2}$

$r = \frac{a_5}{a_3}$

$r = \frac{54}{6}$

r = 9

So, the explicit formula is:

$b_n = b_1 * r^{n-1$

b_1=a_1 , so:

$b_n = a_1 * r^{n-1$

$a_1=\frac{2}{3}$ , so:

$b_n = \frac{2}{3} * 9^{n-1$

Split:

$b_n = \frac{2}{3} * \frac{9^n}{9}$

$b_n = \frac{2}{3*9} (9^n)$

$b_n = \frac{2}{27} (9^n)$

The explicit rule is: $b_n = \frac{2}{27} (9^n)$

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