Question

We live in a connected world where communicating with people is made easier through​ technology, specifically through social networks. With the use of​ Facebook, Twitter, or​ Linked-In, you can talk to a celebrity or someone in another country as easily as you might talk to someone at your work or school. Whether your goal is business or personal​ relationships, it has never been easier to access a multitude of people quickly.
   It has been said that there are only six degrees of separation between any two people in the world. This means that any person can be connected to any other person in the world by 6 or fewer steps. For​ example, you may know someone who knows someone else who knows a person who knows someone who knows a​ celebrity, thereby linking you to the celebrity. Some have said that in our modern era of​ technology, we may even be separated by only 4 or fewer steps.
     Suppose you join an online social network and make 10 friends. Suppose that each of them has 10 friends with whom you yourself are not friends. Suppose each of those friends has 10 friends with whom you are not friends. After these 3​ steps, how many total connections do you​ have? After​ 4, 5, and 6​ steps, how many total connections do you​ have? Write a general equation that models the total number of connections after 6 steps as a function of your initial number of friends. Assume each person has the same initial number of friends. Define any variables used.
     You can reach many people in 6 steps starting with just 10​ friends, but what if you wanted to reach a number of people equal to the population of the whole​ world? How many friends would you need initially​ (and on each​ step) to connect with the number of people that is equal to the size of the entire world population in 6​ steps? Use the model and the known values to write a specific​ equation, and solve it numerically using Excel.
     Now choose a realistic number of​ people, such as the number of people at your school or in your​ town, that you want to reach in 6 steps. How many friends would you need initially​ (and on each​ step) in order to connect with this number of​ people? Use the model and the known values to write a specific​ equation, and solve it numerically using Excel. I want last part only.

Answer 1

I start with 10 friends after 1st step.

After second step , i will have 10*10 firnds because each of the firend from first step is connected to 10 new friends.This will be 100 friends.

After third step, each of the 100 friends from step 2 will have 10 firnds each. Thus overall, i will have 100*10 firnds. This is equal to 1000.

Simlilarly after fourth step , i will have 10000 frinds and so on.

Ideally we can observe that with each step , my firnds are increasing 10 times.

Thus if i had N steps ,then my firnds will be 10000... N times .This can be written as (10)^{N .}

Thus after N steps , total number of firnds will be (10)^{N .}

^{Now if i had X firnds to atart with and each of these X friends have X friends in successive steps then after N steps i will have}

(X)^N friends.

Thus generally i will write that

If i have X friends and each friend has X friend each and it carries on with each step then after N steps i will have (X)^N friends.

with 10 firnds, after 6 steps i will have (10)⁶ friends (a million friends) .

b). Intially friends to cover total world population in 6 steps

Let total number of initial friends be X .

As i am writing this answer, the total world population was approximately 7.7 billions .

Thus For our condition to satisfy

(X)⁶ =7.7*10⁹

From the above expression we can get the value of x

X =44.43 ~44 friends

Thus to befriend entire world population in l6 steps ,i need to start with about 44 friends.

c). Intially friends to cover total school population in 6 steps

Let total number of initial friends be X .

Assume that the total school population is approximately 1000 students.

Then X⁶ =1000

Hence ,X =3.16 ~3 friends