At 10: 00, 5 nc state students learn of a rumor. in the next minute (between 10: 00 and 10: 01), these students each text two of their friends who have not yet heard the rumor. as a result of these texts, 10 new individuals learn of the rumor in the first minute of the rumor's spread. in the next minute, these 10 individuals each text two friends who have not yet heard the rumor. as a result, in the second minute of the rumor's spread 20 new students learn of the rumor. let pn denote the number of students who learn of the rumor in the nth minute of the rumor's spread. assume that the rumor continues to spread according to the pattern described above (with each person who learns of the rumor texting two friends in the minute after they learned the rumor.)

(a) what is p0? find p3.
(b) the sequence p0, p1, p2, p3, is sequence.
(c) how many individuals learned of the rumor in the fifth minute of the rumor's spread?
(d) how many individuals learned of the rumor in the 25th minute of the rumor's spread?
(e) at 10: 25, what is the total number of individuals who have heard the rumor?

Step-by-step explanation:

given that at 10:00, 5 NC State students learn of a rumor. In the next minute (between 10:00 and 10:01), these students each text two of their friends who have not yet heard the rumor. As a result of these texts, 10 new individuals learn of the rumor in the first minute of the rumor's spread. In the next minute, these 10 individuals each text two friends who have not yet heard the rumor. As a result, in the second minute of the rumor's spread 20 new students learn of the rumor.

(a) What is P0? Find P3.

(b) The sequence P0, P1, P2, P3, ... is sequence.

Geometrical

(c) How many individuals learned of the rumor in the fifth minute of the rumor's spread?

(d) How many individuals learned of the rumor in the 25th minute of the rumor's spread?

(e) At 10:25, what is the total number of individuals who have heard the rumor?

At 10.25 from 0 we have 25 minutes have elapsed

Total number = Sum of first 26 terms of GP

=

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))