💡First, I need to calculate the value of (2^{10} - 1). [2^{10} = 1024][2^{10} - 1 = 1024 - 1 = 1023]Next, I will factorize 1023 to find its prime factors. I notice that 1023 is divisible by 3 because the sum of its digits (1 + 0 + 2 + 3 = 6) is divisible by 3.[1023 div 3 = 341]Now, I will factorize 341. I check for divisibility by smaller prime numbers:- 341 is not divisible by 2, 3, or 5.- Checking divisibility by 11:[341 div 11 = 31]Finally, I check if 31 is a prime number. Since 31 is only divisible by 1 and itself, it is a prime number.So, the prime factors of 1023 are 3, 11, and 31. To find the sum of these prime factors:[3 + 11 + 31 = 45]

💡First, I need to calculate (3 * 2) using the defined operation (a * b = a^b - b^a).Calculating (3 * 2):[3^2 = 9][2^3 = 8][3 * 2 = 9 - 8 = 1]Next, I use the result from (3 * 2) to calculate ((3 * 2) * 4), which is (1 * 4).Calculating (1 * 4):[1^4 = 1][4^1 = 4][1 * 4 = 1 - 4 = -3]Therefore, the final answer is (-3).

💡Okay, so I have this problem here: Sam had 98 pennies in his bank. He spent 93 of his pennies. How many pennies does he have now? Alright, let's try to figure this out step by step.First, I need to understand what the problem is asking. Sam starts with a certain number of pennies, which is 98. Then he spends some of them, specifically 93 pennies. The question is asking how many pennies he has left after spending that amount.So, in simple terms, this is a subtraction problem. We have an initial amount, and then we subtract the amount that was spent to find out what's left. Let me write that down:Initial amount: 98 penniesAmount spent: 93 penniesAmount remaining: ?So, the formula would be:Amount remaining = Initial amount - Amount spentPlugging in the numbers:Amount remaining = 98 - 93Now, let's do the subtraction. 98 minus 93. Hmm, 93 plus 5 is 98, right? So, 98 minus 93 should be 5. That seems straightforward.Wait, let me double-check to make sure I didn't make a mistake. Sometimes, when subtracting larger numbers, it's easy to get confused. So, if I have 98 pennies and I take away 93, how many are left?Let's count down from 98:98 - 93 = ?If I subtract 90 from 98, that leaves me with 8 pennies. Then, I still need to subtract the remaining 3 pennies. So, 8 - 3 = 5.Yes, that matches what I got before. So, it looks like the amount remaining is 5 pennies.Just to be extra sure, I'll try adding the amount spent back to the remaining amount to see if it equals the initial amount. That's a good way to verify my answer.So, if Sam has 5 pennies left and he spent 93, then:5 + 93 = 98Which matches the initial amount he had. That means my subtraction was correct.Another way to think about it is to break down the numbers. Let's see:98 can be thought of as 90 + 8.93 can be thought of as 90 + 3.So, subtracting 93 from 98:(90 + 8) - (90 + 3) = (90 - 90) + (8 - 3) = 0 + 5 = 5Again, I get 5 pennies remaining. This method also confirms my previous result.I could also use a number line to visualize this. Starting at 98 and moving back 93 units. Each unit represents one penny. So, from 98, if I move back 93, I land on 5. That's another way to see that the answer is 5.Alternatively, I can think about it in terms of money. If I have 98 cents and I spend 93 cents, how much do I have left? It's the same concept. 98 cents minus 93 cents is 5 cents. So, that reinforces the idea that the answer is 5 pennies.Wait a second, let me make sure I'm not missing anything here. The problem says Sam had 98 pennies and spent 93 of them. Is there any possibility that he could have spent more than he had? Well, 93 is less than 98, so he didn't overspend. That means the result should be a positive number, which it is—5 pennies.Also, considering the context, pennies are individual units, so we're dealing with whole numbers here. There's no mention of fractions or decimals, so the answer should be a whole number, which 5 is.Just to cover all bases, let's consider if the problem had said he spent 98 pennies. Then he would have 0 pennies left. But since he spent 93, which is 5 less than 98, he has 5 pennies remaining.I think I've covered all the angles here. I've done the subtraction multiple ways, checked my work by adding back, broken down the numbers, used a number line, and even thought about it in terms of money. All methods point to the same answer: 5 pennies.So, I'm confident that the correct answer is 5 pennies.

💡Okay, so I have this problem here about sequences. Let me try to understand it step by step. First, they give me the sequence {a_n} with a₁ = 2 and a₂ = 4. Then there's another sequence {b_n} defined by b_n = a_{n+1} - a_n. So, b_n is just the difference between consecutive terms of the a sequence. Got that.Next, they tell me that b_{n+1} = 2b_n + 2. Hmm, so each term of the b sequence is twice the previous term plus 2. That seems like a recursive formula. I need to prove that {b_n + 2} is a geometric sequence. Alright, let me think about how to approach this. Maybe I can manipulate the given recursive formula for b_n to see if adding 2 to both sides helps. Let's try that.Starting with b_{n+1} = 2b_n + 2. If I add 2 to both sides, I get:b_{n+1} + 2 = 2b_n + 2 + 2 b_{n+1} + 2 = 2b_n + 4 Hmm, that doesn't seem immediately helpful. Wait, maybe I can factor out something. Let me see:Looking back at the original equation: b_{n+1} = 2b_n + 2. If I add 2 to both sides, it becomes:b_{n+1} + 2 = 2b_n + 4 But 2b_n + 4 can be written as 2(b_n + 2). So,b_{n+1} + 2 = 2(b_n + 2)Oh, that's nice! So, this shows that each term of the sequence {b_n + 2} is twice the previous term. That means {b_n + 2} is a geometric sequence with common ratio 2. Now, I need to find the first term of this geometric sequence. The first term would be when n = 1. So, b₁ + 2. But what is b₁? Since b_n = a_{n+1} - a_n, then b₁ = a₂ - a₁ = 4 - 2 = 2. So, b₁ + 2 = 2 + 2 = 4. Therefore, the sequence {b_n + 2} is a geometric sequence with the first term 4 and common ratio 2. That proves part (1). Moving on to part (2), I need to find the general term formula for {a_n}. From part (1), we know that {b_n + 2} is a geometric sequence with first term 4 and ratio 2. So, the general term for {b_n + 2} is:b_n + 2 = 4 * 2^{n-1}Simplifying that, since 4 is 2², it becomes:b_n + 2 = 2² * 2^{n-1} = 2^{n+1}Therefore, b_n = 2^{n+1} - 2.So, we have b_n = a_{n+1} - a_n = 2^{n+1} - 2.Now, to find a_n, I can use the fact that a_{n+1} = a_n + b_n. So, starting from a₁, I can write:a₂ = a₁ + b₁ a₃ = a₂ + b₂ a₄ = a₃ + b₃ ... a_n = a_{n-1} + b_{n-1}So, if I sum all these up, I can express a_n in terms of a₁ and the sum of b terms.Let's write that out:a_n = a₁ + (b₁ + b₂ + ... + b_{n-1})We know a₁ = 2, and each b_k = 2^{k+1} - 2. So, let's substitute that in:a_n = 2 + Σ_{k=1}^{n-1} (2^{k+1} - 2)I can split this sum into two separate sums:a_n = 2 + Σ_{k=1}^{n-1} 2^{k+1} - Σ_{k=1}^{n-1} 2First, let's compute Σ_{k=1}^{n-1} 2^{k+1}. That's the same as 2² + 2³ + ... + 2^{n}. This is a geometric series with first term 2² = 4, ratio 2, and number of terms (n - 1). The sum of a geometric series is S = a₁(r^{m} - 1)/(r - 1), where m is the number of terms.So, S₁ = 4*(2^{n - 1} - 1)/(2 - 1) = 4*(2^{n - 1} - 1) = 2^{n + 1} - 4.Next, the second sum is Σ_{k=1}^{n-1} 2, which is just 2*(n - 1).Putting it all together:a_n = 2 + (2^{n + 1} - 4) - 2*(n - 1)Simplify this:a_n = 2 + 2^{n + 1} - 4 - 2n + 2 Combine like terms:2 - 4 + 2 = 0 So, a_n = 2^{n + 1} - 2nLet me check this formula with the given terms to make sure it's correct.For n = 1: a₁ = 2^{2} - 2*1 = 4 - 2 = 2. Correct.For n = 2: a₂ = 2^{3} - 2*2 = 8 - 4 = 4. Correct.For n = 3: a₃ = 2^{4} - 2*3 = 16 - 6 = 10. Let's verify using b₂.b₂ = 2^{3} - 2 = 8 - 2 = 6 a₃ = a₂ + b₂ = 4 + 6 = 10. Correct.For n = 4: a₄ = 2^{5} - 2*4 = 32 - 8 = 24 b₃ = 2^{4} - 2 = 16 - 2 = 14 a₄ = a₃ + b₃ = 10 + 14 = 24. Correct.Looks good! So, the general term formula for {a_n} is a_n = 2^{n + 1} - 2n.

💡Okay, so I have this problem here about electronic products where their sizes follow a normal distribution N(1, 4). I need to find the probability that a randomly selected product has a size between 3 and 5. The options are A: 0.3174, B: 0.2718, C: 0.1359, and D: 0.0456.First, let me make sure I understand what N(1, 4) means. In normal distribution notation, N(μ, σ²), so here μ is 1 and σ² is 4. That means the mean size is 1, and the variance is 4. Therefore, the standard deviation σ is the square root of 4, which is 2.So, μ = 1 and σ = 2.Now, I need to find P(3 < X < 5). That's the probability that the size X is between 3 and 5.Since it's a normal distribution, I can standardize this interval to find the corresponding Z-scores and then use the standard normal distribution table or the given probabilities to find the area under the curve between these Z-scores.Let me recall the formula for converting a normal variable X to a standard normal variable Z:Z = (X - μ) / σSo, for X = 3:Z1 = (3 - 1) / 2 = 2 / 2 = 1And for X = 5:Z2 = (5 - 1) / 2 = 4 / 2 = 2So, now I need to find P(1 < Z < 2), where Z is the standard normal variable.I remember that the total area under the standard normal curve is 1, and the probabilities for certain ranges are given. The note in the problem says:- P(μ - σ < X < μ + σ) = 0.6827- P(μ - 2σ < X < μ + 2σ) = 0.9545So, in terms of Z-scores, since μ = 0 and σ = 1 for the standard normal distribution, these correspond to:- P(-1 < Z < 1) = 0.6827- P(-2 < Z < 2) = 0.9545But in our case, we need P(1 < Z < 2). Hmm, so that's the area between Z = 1 and Z = 2.I think the way to find this is to subtract the cumulative probability up to Z = 1 from the cumulative probability up to Z = 2.In other words:P(1 < Z < 2) = P(Z < 2) - P(Z < 1)From the standard normal distribution table, I know that:- P(Z < 1) ≈ 0.8413- P(Z < 2) ≈ 0.9772So, subtracting these:P(1 < Z < 2) = 0.9772 - 0.8413 = 0.1359Wait, that's one of the options. Option C is 0.1359.But let me double-check if I did everything correctly.First, I converted X to Z correctly:For X = 3: Z = (3 - 1)/2 = 1For X = 5: Z = (5 - 1)/2 = 2So, yes, the interval (3, 5) corresponds to (1, 2) in Z-scores.Then, using the standard normal probabilities:P(Z < 2) is about 0.9772, and P(Z < 1) is about 0.8413.Subtracting gives 0.1359, which is option C.Alternatively, using the given note in the problem, which provides P(μ - σ < X < μ + σ) = 0.6827 and P(μ - 2σ < X < μ + 2σ) = 0.9545.But in our case, the interval is (3,5). Let's see how this relates to μ and σ.Given μ = 1 and σ = 2, so:μ + σ = 1 + 2 = 3μ + 2σ = 1 + 4 = 5So, the interval (3,5) is exactly (μ + σ, μ + 2σ).Therefore, the probability P(3 < X < 5) is the same as P(μ + σ < X < μ + 2σ).From the note, we know that P(μ - 2σ < X < μ + 2σ) = 0.9545, and P(μ - σ < X < μ + σ) = 0.6827.Since the normal distribution is symmetric, the area between μ + σ and μ + 2σ is the same as the area between μ - 2σ and μ - σ.Therefore, the total area between μ + σ and μ + 2σ is (0.9545 - 0.6827)/2.Calculating that:0.9545 - 0.6827 = 0.2718Divide by 2: 0.2718 / 2 = 0.1359So, that's the same result as before, 0.1359, which is option C.Therefore, I can be confident that the correct answer is C: 0.1359.**Final Answer**boxed{C}

💡Okay, so I have this triangle ABC, and I need to solve two parts. First, find the value of cos A, and second, if a is 4, find the value of c. Let me start by understanding the given information.It says that in triangle ABC, the sides opposite angles A, B, and C are a, b, and c respectively. Also, it's given that cos C is 1/8 and that angle C is twice angle A. So, C = 2A. Starting with part (1): Find cos A.Hmm, okay. Since C is twice A, maybe I can use some trigonometric identities here. I remember that cos(2θ) can be expressed in terms of cos θ. Specifically, cos(2θ) = 2cos²θ - 1. So, if I let θ be angle A, then cos C = cos(2A) = 2cos²A - 1.Given that cos C is 1/8, I can set up the equation:2cos²A - 1 = 1/8Let me solve for cos²A. Adding 1 to both sides:2cos²A = 1 + 1/8 = 9/8Then, divide both sides by 2:cos²A = 9/16So, cos A is the square root of 9/16, which is 3/4. But wait, do I need to consider the negative root? Since in a triangle, all angles are between 0 and 180 degrees, and cos is positive in the first and fourth quadrants. But since angles in a triangle are between 0 and 180, and if angle C is 2A, then angle A must be less than 90 degrees because C is 2A, and the sum of angles in a triangle is 180. So, if C is 2A, and A is less than 90, then C is less than 180, which is fine. So, cos A is positive, so cos A is 3/4.Okay, that seems straightforward. So, part (1) is 3/4.Moving on to part (2): If a = 4, find c.Alright, so a is the side opposite angle A, which is 4. We need to find c, which is opposite angle C.I think the Law of Sines would be helpful here. The Law of Sines states that a/sin A = b/sin B = c/sin C.So, if I can find sin A and sin C, I can set up the ratio and solve for c.First, let's find sin A. Since cos A is 3/4, we can use the identity sin²A + cos²A = 1.So, sin²A = 1 - cos²A = 1 - (9/16) = 7/16Therefore, sin A = sqrt(7/16) = sqrt7 / 4Similarly, let's find sin C. Since cos C is 1/8, we can use the same identity.sin²C = 1 - cos²C = 1 - (1/64) = 63/64So, sin C = sqrt(63/64) = (3*sqrt7)/8Wait, sqrt(63) is 3*sqrt7, because 63 is 9*7, so sqrt(63) is 3*sqrt7. So, sin C is 3*sqrt7 /8.Now, applying the Law of Sines:a / sin A = c / sin CWe have a = 4, sin A = sqrt7 /4, sin C = 3*sqrt7 /8So, plugging in:4 / (sqrt7 /4) = c / (3*sqrt7 /8)Let me compute the left side first:4 divided by (sqrt7 /4) is 4 * (4 / sqrt7) = 16 / sqrt7So, 16 / sqrt7 = c / (3*sqrt7 /8)Now, solve for c:Multiply both sides by (3*sqrt7 /8):c = (16 / sqrt7) * (3*sqrt7 /8)Simplify this:16 and 8 can be simplified: 16/8 = 2sqrt7 in numerator and denominator cancels out.So, 2 * 3 = 6Therefore, c = 6.Wait, let me double-check that calculation.So, 4 divided by (sqrt7 /4) is indeed 16 / sqrt7.Then, c is equal to (16 / sqrt7) multiplied by (3*sqrt7 /8).Multiplying numerator: 16 * 3*sqrt7 = 48*sqrt7Denominator: sqrt7 *8 = 8*sqrt7So, 48*sqrt7 / 8*sqrt7 = (48/8)*(sqrt7 / sqrt7) = 6*1 = 6Yes, that's correct.So, c is 6.Wait, let me think if there's another way to approach this, just to confirm.Alternatively, since C = 2A, maybe I can find angle B as well, using the fact that the sum of angles in a triangle is 180 degrees.So, A + B + C = 180But C = 2A, so:A + B + 2A = 180So, 3A + B = 180Therefore, B = 180 - 3ABut I don't know if that helps me directly, unless I want to use the Law of Cosines or something else. But since I already have the Law of Sines approach, and it gave me a concrete answer, I think that's solid.Alternatively, maybe using the Law of Cosines on angle C.Given that cos C = 1/8, and C is opposite side c, which is what we're trying to find.Law of Cosines says:c² = a² + b² - 2ab cos CBut wait, I don't know b, so that might not be helpful unless I can express b in terms of a.Alternatively, maybe express b using the Law of Sines as well.Wait, if I use the Law of Sines, I can express b in terms of sin B, but since I don't know angle B, maybe it's more complicated.Alternatively, since I have angle C = 2A, and cos C = 1/8, and I found cos A = 3/4, maybe I can find angle A and angle C in degrees, then angle B, and then use Law of Sines.But since I already have sin A and sin C, and the ratio, I think the previous method is sufficient.So, I think c is indeed 6.Therefore, summarizing:(1) cos A = 3/4(2) c = 6**Final Answer**(1) boxed{dfrac{3}{4}} (2) boxed{6}