answer:C) Bleeder current

answer:The given series is an infinite geometric series with a first term a = frac{35}{36} and a common ratio r = frac{1}{10}. For a geometric series to converge, |r| < 1, which is true in this case since |frac{1}{10}| = frac{1}{10} < 1. The sum of an infinite geometric series is given by the formula S = frac{a}{1 - r}. Applying this formula, we get: S = frac{frac{35}{36}}{1 - frac{1}{10}} S = frac{frac{35}{36}}{frac{9}{10}} S = frac{35}{36} times frac{10}{9} S = frac{350}{324} S = frac{175}{162} So, the sum of the infinite geometric series is frac{175}{162}.

answer:The answer to this problem depends on the parity of the number of seconds, n. It is not possible for the grasshoppers to return to their initial positions if n is odd. However, if n is even, it is possible. Here's a mathematical approach to prove this: Let g_1, g_2, g_3 denote the positions of the three grasshoppers. Consider the determinant of the vectors (g_2 - g_1) and (g_3 - g_1). This determinant represents the signed area of the triangle formed by the grasshoppers' positions. After each hop, the determinant changes sign, which indicates a change in the orientation of the triangle. If the grasshoppers start in a counterclockwise orientation around the triangle's boundary, they will be in a clockwise orientation after an odd number of hops. Since the initial and final orientations must match for the grasshoppers to return to their starting positions, and the orientation changes with each hop, an even number of hops is required for the orientations to align again. Thus, the grasshoppers can only return to their initial positions after an even number of seconds (n is even).

answer:First, we can simplify the left-hand side of the equation using the rule for multiplying powers with the same base: 5^{23 x-19}5^{20 x-21} = 5^{23 x-19 + 20 x-21} = 5^{43 x-40} Next, we can rewrite the right-hand side of the equation in terms of 5, since 25 = 5^2: 25^{17 x-1} = (5^2)^{17 x-1} = 5^{34 x-2} Now we can set the two sides of the equation equal to each other: 5^{43 x-40} = 5^{34 x-2} Since the bases are the same, we can equate the exponents: 43 x-40 = 34 x-2 Simplifying this equation, we get: 9 x = 38 Dividing both sides by 9, we find: x = frac{38}{9} Therefore, the only real solution to the equation 5^{23 x-19}5^{20 x-21} = 25^{17 x-1} is x = frac{38}{9}. The answer is left{xto frac{38}{9}right}