💡First, I recognize that the given sequence ( a_n = dfrac{1}{n(n+1)} ) can be simplified using partial fraction decomposition. By expressing ( a_n ) as ( dfrac{1}{n} - dfrac{1}{n+1} ), the sum ( S_6 ) becomes a telescoping series.Next, I calculate the sum of the first 6 terms by expanding each term:[S_6 = left(1 - dfrac{1}{2}right) + left(dfrac{1}{2} - dfrac{1}{3}right) + left(dfrac{1}{3} - dfrac{1}{4}right) + left(dfrac{1}{4} - dfrac{1}{5}right) + left(dfrac{1}{5} - dfrac{1}{6}right) + left(dfrac{1}{6} - dfrac{1}{7}right)]I observe that most terms cancel out, leaving only the first term of the first fraction and the second term of the last fraction:[S_6 = 1 - dfrac{1}{7} = dfrac{6}{7}]Therefore, the sum of the first 6 terms of the sequence is ( dfrac{6}{7} ).

💡First, I recognize that the retailer initially ordered an equal number of white and yellow tennis balls. Let's denote the original number of white balls as ( W ) and the original number of yellow balls as ( Y ). Therefore, ( W = Y ).However, due to the dispatch clerk's error, 70 extra yellow balls were sent, making the total number of yellow balls ( Y + 70 ). The problem states that the new ratio of white balls to yellow balls is ( frac{8}{13} ). This can be expressed as:[frac{W}{Y + 70} = frac{8}{13}]Since ( W = Y ), I can substitute ( W ) with ( Y ) in the equation:[frac{Y}{Y + 70} = frac{8}{13}]Next, I'll cross-multiply to solve for ( Y ):[13Y = 8(Y + 70)]Expanding the equation:[13Y = 8Y + 560]Subtracting ( 8Y ) from both sides:[5Y = 560]Dividing both sides by 5:[Y = 112]Since the retailer ordered an equal number of white and yellow balls originally, the total number of balls ordered was:[text{Total balls} = W + Y = Y + Y = 112 + 112 = 224]

💡First, I need to determine Milo's running speed based on the information provided about Cory's wheelchair speed.Cory drives his wheelchair at 12 miles per hour, and this speed is twice the speed at which Milo can roll downhill on his skateboard. Therefore, Milo's skateboard speed is half of Cory's wheelchair speed.Once I have Milo's skateboard speed, I know that this speed is twice his running speed. By halving the skateboard speed, I can find Milo's running speed.Finally, to find out how many miles Milo can run in two hours, I multiply his running speed by the time spent running.

💡Okay, so I have this problem that says if the numbers (a, b, c) form a geometric progression, then their logarithms form an arithmetic progression. Hmm, let me try to understand what this means and how to prove it.First, I remember that in a geometric progression, each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if (a, b, c) are in geometric progression, there must be some common ratio (r) such that:[b = a cdot r quad text{and} quad c = b cdot r = a cdot r^2]Got that part. So, (b) is (a) times (r), and (c) is (a) times (r) squared.Now, the problem is asking about the logarithms of these numbers forming an arithmetic progression. I know that in an arithmetic progression, the difference between consecutive terms is constant. So, if (log(a), log(b), log(c)) are in arithmetic progression, then the difference between (log(b)) and (log(a)) should be the same as the difference between (log(c)) and (log(b)).Mathematically, that means:[log(b) - log(a) = log(c) - log(b)]If I can show that this equation holds true, then I've proven that the logarithms form an arithmetic progression.Let me see if I can manipulate the expressions for (b) and (c) to get to this point. Starting with the expressions for (b) and (c):[b = a cdot r quad text{and} quad c = a cdot r^2]If I take the logarithm of each term, I get:[log(b) = log(a cdot r) quad text{and} quad log(c) = log(a cdot r^2)]Using the logarithm property that (log(m cdot n) = log(m) + log(n)), I can rewrite these as:[log(b) = log(a) + log(r) quad text{and} quad log(c) = log(a) + 2log(r)]Okay, so now I have expressions for (log(b)) and (log(c)) in terms of (log(a)) and (log(r)). Let's compute the differences:[log(b) - log(a) = [log(a) + log(r)] - log(a) = log(r)][log(c) - log(b) = [log(a) + 2log(r)] - [log(a) + log(r)] = log(r)]So, both differences are equal to (log(r)), which means:[log(b) - log(a) = log(c) - log(b)]This shows that the logarithms of (a, b, c) have a constant difference, which is the definition of an arithmetic progression. Therefore, if (a, b, c) are in geometric progression, their logarithms are in arithmetic progression.Wait, let me double-check my steps to make sure I didn't make any mistakes. I started with the definitions of a geometric progression, expressed (b) and (c) in terms of (a) and (r), took logarithms, and used logarithm properties to simplify. Then, I calculated the differences between consecutive logarithms and found they were equal. That seems correct.I think another way to look at it is by considering the property of geometric progressions that the square of the middle term is equal to the product of the first and third terms. So, (b^2 = a cdot c). If I take the logarithm of both sides:[log(b^2) = log(a cdot c)]Using logarithm properties:[2log(b) = log(a) + log(c)]Dividing both sides by 2:[log(b) = frac{log(a) + log(c)}{2}]This shows that (log(b)) is the average of (log(a)) and (log(c)), which is another way of saying that (log(a), log(b), log(c)) are in arithmetic progression.Yeah, that makes sense. So, both approaches confirm that the logarithms form an arithmetic progression if the original numbers form a geometric progression. I think I've got it.**Final Answer**boxed{log(a), log(b), log(c) text{ form an arithmetic progression.}}

💡First, I need to find the slope of the original line defined by the equation (4x - 3y = 12). To do this, I'll convert the equation into slope-intercept form ((y = mx + b)).Starting with the original equation:[4x - 3y = 12]I'll isolate (y) by subtracting (4x) from both sides:[-3y = -4x + 12]Next, I'll divide every term by (-3) to solve for (y):[y = frac{4}{3}x - 4]From this equation, I can see that the slope ((m)) of the original line is (frac{4}{3}).Since the line we're looking for is perpendicular to this original line, its slope will be the negative reciprocal of (frac{4}{3}). The negative reciprocal is calculated by flipping the fraction and changing its sign:[text{Slope of perpendicular line} = -frac{3}{4}]Now that I have the slope of the perpendicular line and know that its (y)-intercept is 4, I can write its equation in slope-intercept form:[y = -frac{3}{4}x + 4]To find the (x)-intercept, I'll set (y = 0) and solve for (x):[0 = -frac{3}{4}x + 4]Subtracting 4 from both sides:[-frac{3}{4}x = -4]Multiplying both sides by (-1) to make the equation positive:[frac{3}{4}x = 4]Finally, I'll solve for (x) by multiplying both sides by the reciprocal of (frac{3}{4}), which is (frac{4}{3}):[x = 4 times frac{4}{3} = frac{16}{3}]Therefore, the (x)-intercept of the line is (frac{16}{3}).

💡First, I need to calculate the sum of the first three numbers: 57, 13, and 72. Adding these together gives me 142.Next, I'll represent the fourth number as a two-digit number, which I'll denote as 10a + b, where 'a' is the tens digit and 'b' is the units digit. This means the total sum of all four numbers will be 142 + 10a + b.Now, I'll calculate the sum of the digits of the first three numbers. Breaking them down:- 57 has digits 5 and 7, which sum to 12.- 13 has digits 1 and 3, which sum to 4.- 72 has digits 7 and 2, which sum to 9.Adding these together, the sum of the digits of the first three numbers is 12 + 4 + 9 = 25.Including the fourth number, the total sum of the digits becomes 25 + a + b.According to the problem, the sum of the digits of all four numbers equals one-fifth of the sum of all four numbers. This gives me the equation:142 + 10a + b = 5(25 + a + b)Simplifying this equation:142 + 10a + b = 125 + 5a + 5bSubtracting 125 + 5a + 5b from both sides:17 + 5a = 4bNow, I need to find the smallest possible value for the fourth number, which means I should look for the smallest possible values of 'a' and 'b' that satisfy this equation. Since 'a' and 'b' are digits, they must be between 0 and 9.Starting with the smallest possible value for 'a', which is 0:17 + 5(0) = 4b17 = 4bb = 17/4 = 4.25Since 'b' must be an integer, this doesn't work.Next, trying 'a' = 1:17 + 5(1) = 4b22 = 4bb = 22/4 = 5.5Again, not an integer.Trying 'a' = 2:17 + 5(2) = 4b27 = 4bb = 27/4 = 6.75Still not an integer.Trying 'a' = 3:17 + 5(3) = 4b32 = 4bb = 32/4 = 8This works because 'b' is an integer.So, when 'a' = 3 and 'b' = 8, the fourth number is 10(3) + 8 = 38.To ensure this is the smallest possible, I'll check if there's a smaller 'a' that could work with a higher 'b' within the digit constraints, but since 'b' must be less than or equal to 9, and 'a' is already at its minimum value that satisfies the equation, 38 is indeed the smallest possible fourth number.