Inductive Reasoning

T

troublemaker676

Junior Member
Joined
Sep 1, 2005
Messages
84
Have a question about a tricky word problem,

At 5:00 am 1 person called two other people to tell them about a huge secret, completeting these calls took 5 minutes. These people called two other people. (Those calls took place at 5:05.) Those people continued the chain by calling two people each. Assuming no one was called twice and that each set of calls occurred exactly 5 minutes after the previous calls, how amny people were aware of the secret at 6:00 am?

Now I set up a table,

Time

5:00am 5:05 5:10 5:15 6:00

# of people who know

3 7 15 31 ?


Then I found the differences of the number of people,

which were: 4,8,and 16

Since the differences were not the same i found the second differences,

which were: all numbers were being multiplied by 2

now i can try and find an equation for the number of people by factoring out the number of people, such as 1,3 for 3 or 1,7 for 7. I can't find a linear pattern from the factors so i kept multiplying the number of people by 2 then 3 then 4 and 6 and none of the factors of these new numbers bring me to a linear pattern. Can someone show me what to do?
 
If A<SUB>n</SUB> is the number of people who know after n 5-min periods, then:
A<SUB>0</SUB>=1, A<SUB>1</SUB>=3 and A<SUB>n</SUB>= A<SUB>n−1</SUB>+2<SUP>n</SUP>.
 
The equation I have is (not sure how to type exponents) 2^n+2 -1, this equation does work and the answer is 16383 people at 6:00 am, but I'm still not sure how to get that equation.
 
The closed form is A<SUB>n</SUB>=2<SUP>n+1</SUP>−1.
 
Take an example.
A<SUB>4</SUB>= A<SUB>3</SUB>+2<SUP>4</SUP>
A<SUB>4</SUB>= A<SUB>2</SUB>+2<SUP>3</SUP>+2<SUP>4</SUP>
A<SUB>4</SUB>= A<SUB>1</SUB>+2<SUP>2</SUP>+2<SUP>3</SUP>+2<SUP>4</SUP>
So on until you get A<SUB>4</SUB>= 2<SUP>0</SUP>+2<SUP>1</SUP>+2<SUP>2</SUP>+2<SUP>3</SUP>+2<SUP>4</SUP>.
So you see that A<SUB>n</SUB>= 2<SUP>0</SUP>+2<SUP>1</SUP>+…+2<SUP>n</SUP>.
A<SUB>n</SUB>=2 A<SUB>n</SUB>−A<SUB>n</SUB>=2<SUP>n+1</SUP>−1
 
I get how you got my equation from yours, however how did you get that equation from the original problem. I got the right equation from my teacher but I'm not sure how he got it. Can you show me. Thanks!
 
After every n 5-min periods, we add 2<SUP>n</SUP> more people to the A<SUB>n−1</SUB> people we already have.
 
I really appreciate your help, but i don't think i am understanding. I was taught to find an equation based on the differences of the first numbers:

3 7 15 31 63

4 8 16 32

2 2 2

when the first differences are not the same you look for the second differences which are multiply by 2, now that the differeces are the same you can try and find an equation by factoring out the first numbers:

1 1 3
3 7 15

but when i do that the differences are not the same, so i tryed to multiply the first numbers by 2,3,4and 6 to see if those factors would work and they didn't. If someone knows how to find an equation from that method and can show me what i did wrong. Thanks.
 
Have a question about a tricky word problem,

At 5:00 am 1 person called two other people to tell them about a huge secret, completeting these calls took 5 minutes. These people called two other people. (Those calls took place at 5:05.) Those people continued the chain by calling two people each. Assuming no one was called twice and that each set of calls occurred exactly 5 minutes after the previous calls, how amny people were aware of the secret at 6:00 am?

How many additional people learmed of the secret every 5 minutes?

...n......1.......2......3.......4......5........6............................13
........5:00..5:05..5:10..5:15..5:20...5:25........................6:00
N........1.......2......4.......8......16......32.........................4096

The sequence of additional people learning the secret every 5 minutes form a geometric progression.

The sum of such a progression is S = a(r^(n-1))/(r-1) where S = the sum, a = the first term, r = the comon factor and n = the number of terms.

With a = 1, r = 2, and n = 13, S = 1(40962^(13-1))/(2-1) = 4096
 
Your table looks much different from the correct one i was given

n..............0.......1.........2.........3..........4
time.......5:00...5:05.....5:10.....5:15.....5:20
#people....3........7.........15........31........63
who know

I know this is the right table, now just finding an equation to find the how many people knew at 6:00am. The right answer is 16383. Still not sure how to get there.
 
I think this might help explain it more,

n..............0.......1.........2.........3..........4
time.......5:00...5:05.....5:10.....5:15.....5:20
#people....3........7.........15........31........63
who know

Diff..............4..........8...........16.........32
Diff...................4............8.............16
Diff..........................2..............2

Once all the differences are the same, here it is the third differences, you can begin to find an equation by again factoring out the # of people

...........1............1..........3
...........3.............7..........5

Then you compare the nth term to those factors to get an equation, but only if those factors have differences that are all the same. But I can't get those differences to be all the same. That is what I really need help doing.
 
My apologies. I used the incorrect expression for the sum of a geometric progression which produced the wrong answer.

To repeat the solution.

Have a question about a tricky word problem,

At 5:00 am 1 person called two other people to tell them about a huge secret, completeting these calls took 5 minutes. These people called two other people. (Those calls took place at 5:05.) Those people continued the chain by calling two people each. Assuming no one was called twice and that each set of calls occurred exactly 5 minutes after the previous calls, how amny people were aware of the secret at 6:00 am?

It appears that there are two interpretaions of this problem. It depends on what the statement "These people called two other people." These people could mean the 2 new people that just learned of the secret or all three people that knew the secret at 5:05. My interpretaion is that only the new people that just learned of the secret each call 2 other people.
Therefore:

At 5:00 am one person knew the secret.
At 5:00 am this 1 person called 2 other people to tell them about the secret, taking 5 minutes to complete the call..
At 5:05, these 2 new people who just learned of the secret, and the original person make a total of 3 that knew the secret.
At 5:05, each of these 2 new people called two other people, starting at 5:05 and taking 5 minutes to complete.
At 5:10, 4 new people had learned of the secret making a total of 7 people knowing the secret.
Those people continued the chain tby calling 2 people each.
Assuming no one was called twice and that each set of calls occurred exactly 5 minutes after the previous calls, how amny people were aware of the secret at 6:00 am?

How many additional people learmed of the secret every 5 minutes?

1 calls 2.ending at 5:05
2 call 4 ending at 5:10
4 call 8
8 call 16
16 call 32 etc. creating

...n......1.......2......3.......4......5........6............................13
........5:00..5:05..5:10..5:15..5:20...5:25.........................6:00
N........1.......2......4.......8......16......32.........................4096 (the number of people learning the secret during each 5 minute time period)

The sequence of additional people learning the secret every 5 minutes form a geometric progression.

The sum of such a progression is S = a(r^n - 1))/(r-1) where S = the sum, a = the first term, r = the comon factor and n = the number of terms.

With a = 1, r = 2, and n = 13, S = 1[(2^(14)) - 1]/(2-1) = 8191


Your table
n..............0.......1.........2.........3..........4
time.......5:00...5:05.....5:10.....5:15.....5:20
#people....3........7.........15........31........63
who know

Your table identifies the total number of people knowing the secret at the end of each 5 minute time period but it is incorrect. At time zero, 5:00 am,1 person knew the secret. At 5:05, 2 additional peopls knew the secret making 3 in all knowing the secret. At 5:10, 4 additional people knew the secret making 7 in all knowing the secret. Carrying this further, your table should look like

n..............0........1........2.........3..........4..........5..........6.........7.........8.........9..........10..........11.........12.
time.......5:00...5:05.....5:10.....5:15.....5:20 ....5:25.....5:30.....5:35....5:40.....5:45.......5:50.......5:55......6:00
#people....1........3........7........15.........31.........63......127.....255......511.....1023.......2047.....4095......8191
who know

Using the geometric progression approach above, where the number of new people learning the secret each 5 minute period are listed, the total number of people knowing the secret at 6:00 am derives from S = a(r^n - 1)/(r - 1).
 
Andrew, the seems to be some discrepancy in the way you are setting up the table.
5:00 am 1 person (knows the secret) . Then “These people called two other people.” Those calls took place at 5:05. (this means that at 5:05 3 people know the secrete). So at 5:10 7 people know, etc. Look at the table.
secrets1jo.gif

TchrWill, you are using a different index than Andrew began using. He uses n=0 for 5:00.

Andrew, Which is it? At 5:00 how many people know, 1 or 3? From you first post it has to be 1. But you begin you table at 3.
If it is 1 at 5:00 then at 6:00 there are 8191 people who know. N=12, 5*12=60.
If it is 3 at 5:00 then 16,383.
 
I'm am getting this table from my teacher, who i assumed was correct, however after re-reading the problem it has to be 1 person at 5:00am and 3 at 5:05 because it took 5 minutes to make the calls, which would make the correct answer as you said 8191. So i am now assuming he has made a mistake.
 
pka,

The problem statement was "At 5:00 am 1 person called two other people" to tell them about a huge secret, completeting these calls took 5 minutes.

Therefore, the first number in the sequnce is 1 not 3 as the proposer mistakenly indicated.

1, 3, 7, 15, 31, etc. is the only logical sequence addressing the statement as made making 8191 the total number knowing the secret at 6:00 am.
 
“first number in the sequnce is 1 not 3 as the proposer mistakenly indicated.”
Exactly right. But the point is the index is 0 not 1.
That is A<SUB>0</SUB>=1 and A<SUB>1</SUB>=3.
The debate is about the index and not the terms of the sequence.
 
You must log in or register to reply here.
Top