Use the formula [tex]S_n=a_1\left(\frac{1-r^n}{1-r}\right)[/tex] to determine the number of black squares in the first five figures of the Sierpinski Carpet given by the partial sum:
[tex]\sum_{k=1}^5(8)^{k-1}[/tex]
For this sum:
[tex]a_1=[/tex] [tex]\square[/tex] [tex]r=[/tex] [tex]\square[/tex] [tex]n=[/tex] [tex]\square[/tex]
Answer (2)
Expand the sum: ∑ k = 1 5 ( 8 ) k − 1 = 8 0 + 8 1 + 8 2 + 8 3 + 8 4 .
Identify the parameters: a 1 = 1 , r = 8 , n = 5 .
Substitute the values into the formula: S 5 = 1 × ( 1 − 8 1 − 8 5 ) .
Calculate the sum: S 5 = 4681 . The values are a 1 = 1 , r = 8 , and n = 5 , and the sum is 4681 .
Explanation
Understanding the Problem We are given the formula for the sum of a geometric series: S_n=a_1\t\tleft(\frac{1-r^n}{1-r}\right) , and we need to determine the values of a 1 , r , and n for the given sum ∑ k = 1 5 ( 8 ) k − 1 . This sum represents the number of black squares in the first five figures of the Sierpinski Carpet.
Expanding the Sum First, let's rewrite the sum ∑ k = 1 5 ( 8 ) k − 1 by expanding it: 8 0 + 8 1 + 8 2 + 8 3 + 8 4 .
Identifying the Parameters Now, we can identify the parameters for the geometric series formula:
a 1 is the first term of the series, which is 8 0 = 1 .
r is the common ratio between consecutive terms, which is 8 .
n is the number of terms in the series, which is 5 .
Stating the Values So we have a 1 = 1 , r = 8 , and n = 5 .
Calculating the Sum Now, we substitute these values into the formula S n = a 1 ( 1 − r 1 − r n ) :
S 5 = 1 × ( 1 − 8 1 − 8 5 ) = − 7 1 − 32768 = − 7 − 32767 = 4681
Final Answer Therefore, the number of black squares in the first five figures of the Sierpinski Carpet is 4681.
Examples
Geometric series can be used to model various real-world phenomena, such as the spread of diseases, the decay of radioactive substances, and the calculation of compound interest. For example, if a disease spreads such that each infected person infects 8 others, the total number of infected people after 5 generations can be calculated using a geometric series with a 1 = 1 , r = 8 , and n = 5 . This is similar to the Sierpinski Carpet problem, where each iteration increases the number of black squares by a factor of 8.
Using the formula for the sum of a geometric series, we find the values a 1 = 1 , r = 8 , and n = 5 . Substituting these into the formula gives us a total number of black squares in the first five figures of the Sierpinski Carpet as 4681. Thus, the answer is 4681.
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