6. 3/10 points | Previous Answers My Notes Ask Your Teacher At 10:00, 5 NC State

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6. 3/10 points | Previous Answers My Notes Ask Your Teacher 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 Pw 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 Po? Find P3 75 (b) The sequence Po, Ps, P2, P3, is a geometicsequence. 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?

Explanation / Answer

In this question each student text two of his friend who have not heard the rumor.

(b).Hence,as we increase every minutes we get a pattern that is geometric progression.

Po P1 P2 P3 P4 p5 . . . . . . . .

5 10 20 40 80 160 . . . . . . .

T1 T2 T3 T4 T5 T6 . . . . . . .

(a).In this P0 (at 10:00) means T1(first term of gp)

So P0 (at 10:00) is 5

P3 (T4)= 40

(c).In fifth minutes P5 (T6) is 160

In this G.P first term T1 (a)=5

Common ratio(r)=T2/T1= 10/5 = 2

nth term in G.P (Tn)= a r^n-1

(d).Students learned rumor in 25th minute P25 means 26th term of G.P T26= 5 . 2^26-1= 5× 2^25 = 5×33554432 =167772160

(e).at 10:25 total no. of students who have heard the rumor.

at 10:25 total no. of term in G.P is 26

Hence, we have to find sum of 26 terms (S26)

In G.P Sn= a (r^n -1)/r-1

S26 = 5 (2^26-1 )/2-1

S26= 5 × 67108863= 335544315 are the total no. of students who have heard the rumor at 10:25.

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