💡Okay, so I have this problem here where I need to find constants a, b, and c such that the equation 1(n² - 1²) + 2(n² - 2²) + ... + n(n² - n²) = a n⁴ + b n² + c holds for all positive integers n. If such constants exist, I need to find them; if not, I have to explain why. Hmm, let me try to figure this out step by step.First, I should probably try to understand what the left side of the equation is. It looks like a sum where each term is k multiplied by (n² - k²), right? So, for each k from 1 to n, we're adding up k(n² - k²). Let me write that out more formally:Sum from k=1 to n of [k(n² - k²)].So, that's the same as Sum from k=1 to n of (k n² - k³). Maybe I can split this into two separate sums:Sum from k=1 to n of k n² - Sum from k=1 to n of k³.Which simplifies to n² Sum from k=1 to n of k - Sum from k=1 to n of k³.Okay, I remember that the sum of the first n natural numbers is given by the formula n(n + 1)/2. And the sum of the cubes of the first n natural numbers is [n(n + 1)/2]². Let me write that down:First sum: n² * [n(n + 1)/2] = n² * (n² + n)/2 = (n⁴ + n³)/2.Second sum: [n(n + 1)/2]² = (n²(n + 1)²)/4 = (n⁴ + 2n³ + n²)/4.So, subtracting the second sum from the first sum:(n⁴ + n³)/2 - (n⁴ + 2n³ + n²)/4.To subtract these, I need a common denominator, which is 4:[2(n⁴ + n³) - (n⁴ + 2n³ + n²)] / 4.Let me expand the numerator:2n⁴ + 2n³ - n⁴ - 2n³ - n².Combine like terms:(2n⁴ - n⁴) + (2n³ - 2n³) + (-n²) = n⁴ + 0 - n² = n⁴ - n².So, the entire expression becomes (n⁴ - n²)/4.Therefore, the original sum simplifies to (n⁴ - n²)/4.So, comparing this to the right side of the equation given in the problem, which is a n⁴ + b n² + c, we can set up the equation:(n⁴ - n²)/4 = a n⁴ + b n² + c.To find a, b, and c, we can equate the coefficients of corresponding powers of n on both sides.Let me rewrite the left side:(1/4) n⁴ + (-1/4) n² + 0.So, comparing term by term:For n⁴: 1/4 = a.For n²: -1/4 = b.For the constant term: 0 = c.So, that gives us a = 1/4, b = -1/4, and c = 0.Wait, but before I conclude, let me verify this with a specific value of n to make sure I didn't make a mistake.Let's take n = 1.Left side: 1(1² - 1²) = 1(1 - 1) = 0.Right side: (1/4)(1)^4 + (-1/4)(1)^2 + 0 = 1/4 - 1/4 + 0 = 0. Okay, that matches.How about n = 2.Left side: 1(4 - 1) + 2(4 - 4) = 1*3 + 2*0 = 3 + 0 = 3.Right side: (1/4)(16) + (-1/4)(4) + 0 = 4 - 1 + 0 = 3. Perfect.Let me try n = 3.Left side: 1(9 - 1) + 2(9 - 4) + 3(9 - 9) = 1*8 + 2*5 + 3*0 = 8 + 10 + 0 = 18.Right side: (1/4)(81) + (-1/4)(9) + 0 = 81/4 - 9/4 = (81 - 9)/4 = 72/4 = 18. Nice, that works too.So, it seems like my values for a, b, and c are correct. Therefore, the constants are a = 1/4, b = -1/4, and c = 0.I think that's it. I don't see any mistakes in my calculations, and the examples I tried worked out. So, yeah, these constants should satisfy the equation for all positive integers n.**Final Answer**The constants are boxed{a = dfrac{1}{4}}, boxed{b = -dfrac{1}{4}}, and boxed{c = 0}.**Final Answer**The constants are boxed{a = dfrac{1}{4}}, boxed{b = -dfrac{1}{4}}, and boxed{c = 0}.

💡Okay, so I have this problem where I need to find the slope of a line that passes through the point (-3, 0) and is tangent to the circle defined by the equation x² + y² = 1. The options given are A: ±1/4, B: ±√2/4, C: ±1/3, and D: ±√3/3. Hmm, let me think about how to approach this.First, I remember that the circle x² + y² = 1 is centered at the origin (0,0) with a radius of 1. The line we're looking for passes through (-3, 0) and is tangent to this circle. Since it's tangent, it should touch the circle at exactly one point.I think the general equation of a line can be written as y = mx + b, where m is the slope and b is the y-intercept. But since the line passes through (-3, 0), I can use this point to find a relationship between m and b. Plugging in x = -3 and y = 0 into the equation gives 0 = m*(-3) + b, which simplifies to b = 3m. So, the equation of the line can be rewritten as y = m(x + 3).Now, since this line is tangent to the circle, the distance from the center of the circle (which is at (0,0)) to the line must be equal to the radius of the circle, which is 1. The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). Let me rewrite the equation of the line in the standard form Ax + By + C = 0. Starting from y = m(x + 3), I can rearrange it to mx - y + 3m = 0. So, A = m, B = -1, and C = 3m.Using the distance formula, the distance from (0,0) to the line is |m*0 + (-1)*0 + 3m| / sqrt(m² + (-1)²) = |3m| / sqrt(m² + 1). Since this distance must equal the radius, which is 1, I can set up the equation |3m| / sqrt(m² + 1) = 1.To solve for m, I can square both sides to eliminate the square root. That gives (9m²) / (m² + 1) = 1. Multiplying both sides by (m² + 1) gives 9m² = m² + 1. Subtracting m² from both sides, I get 8m² = 1. Dividing both sides by 8, I find m² = 1/8. Taking the square root of both sides, m = ±√(1/8) = ±(√2)/4.So, the slopes of the tangent lines are ±√2/4. Looking back at the options, that's option B. Let me double-check my steps to make sure I didn't make any mistakes.1. Wrote the equation of the line passing through (-3, 0) as y = m(x + 3). That seems correct.2. Converted it to standard form: mx - y + 3m = 0. Yes, that's right.3. Applied the distance formula from (0,0) to the line: |3m| / sqrt(m² + 1). That looks correct.4. Set the distance equal to the radius: |3m| / sqrt(m² + 1) = 1. Makes sense.5. Squared both sides: 9m² / (m² + 1) = 1. Correct.6. Solved for m²: 9m² = m² + 1 → 8m² = 1 → m² = 1/8 → m = ±√(1/8) = ±√2/4. All steps seem valid.I think I did everything correctly. So, the answer should be option B: ±√2/4.**Final Answer**The slope of the tangent line is boxed{B}.

💡First, I need to determine the total number of possible 3-word sentences using the four words available in the updated Gnollish language. Since each position in the sentence can be any of the four words, the total number of sentences is 4 multiplied by itself three times, which is 4³ = 64.Next, I need to identify and count the invalid sentences. The rule states that "splargh" cannot come directly before "glumph" or "kreeg." This means I need to look for sequences where "splargh" is immediately followed by either "glumph" or "kreeg."For the first type of invalid sentence, where "splargh" is in the first position and is followed by "glumph," there are 4 choices for the third word. Similarly, for the second type, where "splargh" is in the second position and is followed by "glumph," there are 4 choices for the first word. The same logic applies to sequences where "splargh" is followed by "kreeg," resulting in 4 invalid sentences for each scenario.Adding up all the invalid sentences, I have 4 (splargh glumph _) + 4 (_ splargh glumph) + 4 (splargh kreeg _) + 4 (_ splargh kreeg) = 16 invalid sentences.Finally, to find the number of valid sentences, I subtract the number of invalid sentences from the total number of possible sentences: 64 - 16 = 48.

💡Alright, so I have this problem where I need to find the number of geometric sequences of length 4 that can be formed from the set S = {1, 2, 3, ..., 500}. The sequences need to be in ascending order, and the common ratio has to be a positive integer. Hmm, okay, let me try to break this down step by step.First, I remember that a geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if I denote the first term as 'a' and the common ratio as 'q', then the sequence would look like a, aq, aq², aq³. Since we're dealing with four terms, the last term is aq³.Now, all these terms need to be within the set S, which means each term must be an integer between 1 and 500, inclusive. Also, since we're arranging them in ascending order, 'q' must be greater than 1 because if q were 1, all terms would be equal, which isn't allowed since the numbers must be different. So, q has to be at least 2.Let me think about how to approach this. For each possible common ratio 'q', I can find the maximum possible value of 'a' such that aq³ is still less than or equal to 500. Then, for each 'q', the number of valid sequences would be equal to the number of valid starting terms 'a'.So, the first step is to figure out the possible values of 'q'. Since aq³ ≤ 500 and a has to be at least 1, the maximum value of q³ would be 500. Taking the cube root of 500 gives me approximately 7.937, so q can be at most 7 because q has to be an integer. Therefore, the possible values of q are 2, 3, 4, 5, 6, and 7.Now, for each of these q values, I need to find how many starting terms 'a' are possible such that aq³ ≤ 500. Let's go through each q one by one.Starting with q = 2:- The last term is a*2³ = 8a.- So, 8a ≤ 500 => a ≤ 500/8 = 62.5.- Since a has to be an integer, a can be from 1 to 62. So, there are 62 possible sequences for q = 2.Next, q = 3:- The last term is a*3³ = 27a.- So, 27a ≤ 500 => a ≤ 500/27 ≈ 18.518.- Rounding down, a can be from 1 to 18. So, 18 possible sequences for q = 3.Moving on to q = 4:- The last term is a*4³ = 64a.- So, 64a ≤ 500 => a ≤ 500/64 ≈ 7.8125.- Rounding down, a can be from 1 to 7. So, 7 possible sequences for q = 4.Now, q = 5:- The last term is a*5³ = 125a.- So, 125a ≤ 500 => a ≤ 500/125 = 4.- Therefore, a can be 1, 2, 3, or 4. That's 4 sequences for q = 5.Next, q = 6:- The last term is a*6³ = 216a.- So, 216a ≤ 500 => a ≤ 500/216 ≈ 2.3148.- Rounding down, a can be 1 or 2. So, 2 sequences for q = 6.Finally, q = 7:- The last term is a*7³ = 343a.- So, 343a ≤ 500 => a ≤ 500/343 ≈ 1.457.- Rounding down, a can only be 1. So, 1 sequence for q = 7.Now, adding up all these possible sequences:- q=2: 62- q=3: 18- q=4: 7- q=5: 4- q=6: 2- q=7: 1Total = 62 + 18 + 7 + 4 + 2 + 1 = 94.Wait, let me double-check that addition to make sure I didn't make a mistake. 62 + 18 is 80, plus 7 is 87, plus 4 is 91, plus 2 is 93, plus 1 is 94. Yep, that seems right.So, it looks like there are 94 such geometric sequences. I think that's the answer. But just to be thorough, let me consider if there's any chance I missed a q value or miscalculated the number of a's for each q.For q=2, 8a ≤500, a=62.5, so 62. That seems correct.q=3, 27a ≤500, a≈18.518, so 18. Correct.q=4, 64a ≤500, a≈7.8125, so 7. Correct.q=5, 125a ≤500, a=4. Correct.q=6, 216a ≤500, a≈2.3148, so 2. Correct.q=7, 343a ≤500, a≈1.457, so 1. Correct.And q=8 would give 512a, which is already over 500 even for a=1, so q=8 is too big. So, q only goes up to 7.Therefore, I think 94 is the correct count of such geometric sequences.**Final Answer**The number of such geometric sequences is boxed{94}.

💡Okay, so I've got this equation here: 5^(x+1) * 4^(y-1) = 25^x * 64^y. And I also know that x + y = 0.5. The question is asking for the exponent of 4 on the left side of the equation. Hmm, let's break this down step by step.First, I need to understand what the equation is saying. On the left side, we have 5 raised to the power of (x+1) multiplied by 4 raised to the power of (y-1). On the right side, we have 25 raised to the power of x multiplied by 64 raised to the power of y. I remember that 25 is the same as 5 squared, and 64 is the same as 4 cubed. Maybe I can rewrite the right side using these expressions to make the equation easier to compare. So, 25^x becomes (5^2)^x, and 64^y becomes (4^3)^y. Now, using the power of a power rule, which says that (a^m)^n = a^(m*n), I can simplify the right side further. So, (5^2)^x becomes 5^(2x), and (4^3)^y becomes 4^(3y). Now the equation looks like this: 5^(x+1) * 4^(y-1) = 5^(2x) * 4^(3y). Since the bases on both sides of the equation are the same (5 and 4), I can set the exponents equal to each other. That means for the base 5, x + 1 should equal 2x, and for the base 4, y - 1 should equal 3y. Let's solve for x first. If x + 1 = 2x, then subtracting x from both sides gives me 1 = x. So, x equals 1. Now, I know that x + y = 0.5. Since x is 1, I can plug that into the equation: 1 + y = 0.5. Subtracting 1 from both sides gives me y = -0.5. Now, the question is asking for the exponent of 4 on the left side of the equation. On the left side, the exponent of 4 is (y - 1). Since y is -0.5, I can substitute that in: -0.5 - 1 = -1.5. So, the exponent of 4 on the left side of the equation is -1.5.

💡First, I recognize that the initial cost price (CP) of the article is given as 1000.Next, I calculate the new cost price (NCP) if the article were bought at 5% less:NCP = 1000 - (5/100) * 1000 = 950.Then, I determine the new selling price (NSP) that would yield a 10% profit on the new cost price:NSP = 110% of NCP = 1.1 * 950 = 1045.Since the new selling price is 5 less than the original selling price (SP), I set up the equation:1045 = SP - 5.Solving for SP gives:SP = 1050.Finally, I calculate the initial profit percentage:Profit = SP - CP = 1050 - 1000 = 50.Profit Percentage = (50 / 1000) * 100% = 5%.