1. A teacher teaches 8 students how to fold an origami model. Each of these students goes on to teach 8 students of their own how to fold the same model. If this teaching process goes on for 6 generations, how many people total will know how to fold the origami model?
2. By expressing each representing decimal as the sum of an infinite geometric series, find the common fraction equivalent to each.
a. 0.123 123 123
b. 0.672 727 272
c. 3.242424
#1 think 8^n
#2 .123 123 123 = 123/999
same for the others
Bot, Please answer this question
Find the interquartile range (IQR) of the data
98 132 295 68 176 89 287 198 94 68 240 94 369 94 155
Bot, Please answer this question
Find the interquartile range of the data
98 132 295 68 176 89 287 198 94 68 240 94 369 94 155
1. To find the total number of people who will know how to fold the origami model after 6 generations, we can use the formula for the sum of a geometric series.
In this case, the first generation starts with 8 students who learn how to fold the origami model. Each of these students goes on to teach 8 students of their own, making it 8 * 8 = 64 students in the second generation. This process continues for 6 generations.
To calculate the total number of people, we can use the formula:
Total = a * (r^n - 1) / (r - 1)
Where:
a = the first term in the series (initial number of students)
r = the common ratio (number of students each student teaches)
n = the number of terms (number of generations)
In this case:
a = 8 (first generation)
r = 8 (each student teaches 8 students)
n = 6 (6 generations)
Plugging the values into the formula, we get:
Total = 8 * (8^6 - 1) / (8 - 1)
Total = 8 * (262,144 - 1) / 7
Total = 8 * 262,143 / 7
Total = 2,097,744 / 7
Total = 299,678.2857
Since we can't have a fraction of a person, we need to round down to the nearest whole number.
Therefore, the total number of people who will know how to fold the origami model after 6 generations is 299,678.
2. To express each repeating decimal as the sum of an infinite geometric series and find the common fraction equivalent, we can use the following steps:
a. 0.123 123 123
We observe that the decimal repeats every three digits (123) and can be expressed as a series with a common ratio of 0.001 (since each digit is 1/1000th place value) and a first term of 0.123.
Using the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
Substituting the values:
Sum = 0.123 / (1 - 0.001)
Sum = 0.123 / 0.999
Sum = 0.123123123...
The common fraction equivalent is 0.123123123... which can also be written as 123/999.
b. 0.672 727 272
The decimal repeats every nine digits (272727272) and can be expressed as a series with a common ratio of 0.01 (since each digit is 1/100th place value) and a first term of 0.672.
Sum = a / (1 - r)
Sum = 0.672 / (1 - 0.01)
Sum = 0.672 / 0.99
Sum = 0.678787878...
The common fraction equivalent is 0.678787878... which can be simplified by noticing that it repeats every three digits. So, we can write it as 678/990, which can further be reduced to 226/330.
c. 3.242424
The decimal repeats every two digits (24) and can be expressed as a series with a common ratio of 0.01, and a first term of 3.24.
Sum = a / (1 - r)
Sum = 3.24 / (1 - 0.01)
Sum = 3.24 / 0.99
Sum = 3.27272727...
The common fraction equivalent is 3.27272727... which can be simplified by noticing that it repeats every two digits. So, we can write it as 327/99.